%------------------------------------------------------------------------------
% File     : ITP253_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Bounds 01473_083067
% 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    : 0071_VEBT_Bounds_01473_083067 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12376 (4378 unt;1976 typ;   0 def)
%            Number of atoms       : 19023 (8278 equ)
%            Maximal formula atoms :   39 (   1 avg)
%            Number of connectives : 20724 (2000   ~; 330   |;2398   &)
%                                         (2063 <=>;13933  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   31 (   2 avg)
%            Number of FOOLs       : 1199 ( 555 fml; 644 var)
%            Number of X terms     :  813 (   0  []; 644 ite; 169 let)
%            Number of types       :   13 (  12 usr)
%            Number of type conns  : 1691 (1515   >; 176   *;   0   +;   0  <<)
%            Number of predicates  :  236 ( 233 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1751 (1751 usr;  80 con; 0-7 aty)
%            Number of variables   : 37315 (33060   !; 995   ?;37315   :)
%                                         (3260  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TX1_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 03:54:30.681
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

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

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

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

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

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

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

tff(ty_tf_a,type,
    a: $tType ).

% Explicit typings (1957)
tff(sy_cl_Lattices_Obounded__lattice,type,
    bounded_lattice: 
      !>[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_GCD_Oring__gcd,type,
    ring_gcd: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ominus,type,
    minus: 
      !>[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_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_GCD_Osemiring__Gcd,type,
    semiring_Gcd: 
      !>[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_Lattices_Olattice,type,
    lattice: 
      !>[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_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_Complete__Lattices_OInf,type,
    complete_Inf: 
      !>[A: $tType] : $o ).

tff(sy_cl_Complete__Lattices_OSup,type,
    complete_Sup: 
      !>[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_Fields_Olinordered__field,type,
    linordered_field: 
      !>[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_Lattices_Odistrib__lattice,type,
    distrib_lattice: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_Lattices_Osemilattice__sup,type,
    semilattice_sup: 
      !>[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_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_Enum_Ofinite__distrib__lattice,type,
    finite8700451911770168679attice: 
      !>[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_Topological__Spaces_Ot3__space,type,
    topological_t3_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_Real__Vector__Spaces_Odist__norm,type,
    real_V6936659425649961206t_norm: 
      !>[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_GCD_Osemiring__gcd__mult__normalize,type,
    semiri6843258321239162965malize: 
      !>[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_Complete__Lattices_Ocomplete__lattice,type,
    comple6319245703460814977attice: 
      !>[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_Topological__Spaces_Oopen__uniformity,type,
    topolo569519726778239578ormity: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
    comple5582772986160207858norder: 
      !>[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_Lattices_Obounded__semilattice__inf__top,type,
    bounde4346867609351753570nf_top: 
      !>[A: $tType] : $o ).

tff(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
    bounde4967611905675639751up_bot: 
      !>[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_Complete__Lattices_Ocomplete__boolean__algebra,type,
    comple489889107523837845lgebra: 
      !>[A: $tType] : $o ).

tff(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
    comple592849572758109894attice: 
      !>[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_Conditionally__Complete__Lattices_Olinear__continuum,type,
    condit5016429287641298734tinuum: 
      !>[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_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
    counta3822494911875563373attice: 
      !>[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__lattice,type,
    condit1219197933456340205attice: 
      !>[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] : fun(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] : fun(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] : ( 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] : ( 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] : ( 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] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ann____,type,
    aTP_Lamp_ann: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ano____,type,
    aTP_Lamp_ano: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anp____,type,
    aTP_Lamp_anp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anq____,type,
    aTP_Lamp_anq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anr____,type,
    aTP_Lamp_anr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ans____,type,
    aTP_Lamp_ans: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ant____,type,
    aTP_Lamp_ant: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anu____,type,
    aTP_Lamp_anu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anv____,type,
    aTP_Lamp_anv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anw____,type,
    aTP_Lamp_anw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anx____,type,
    aTP_Lamp_anx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__any____,type,
    aTP_Lamp_any: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anz____,type,
    aTP_Lamp_anz: 
      !>[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__aoa____,type,
    aTP_Lamp_aoa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aob____,type,
    aTP_Lamp_aob: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoc____,type,
    aTP_Lamp_aoc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aod____,type,
    aTP_Lamp_aod: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoe____,type,
    aTP_Lamp_aoe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aof____,type,
    aTP_Lamp_aof: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aog____,type,
    aTP_Lamp_aog: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aoh____,type,
    aTP_Lamp_aoh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoi____,type,
    aTP_Lamp_aoi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoj____,type,
    aTP_Lamp_aoj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aok____,type,
    aTP_Lamp_aok: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aol____,type,
    aTP_Lamp_aol: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aom____,type,
    aTP_Lamp_aom: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aon____,type,
    aTP_Lamp_aon: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoo____,type,
    aTP_Lamp_aoo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aop____,type,
    aTP_Lamp_aop: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoq____,type,
    aTP_Lamp_aoq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aor____,type,
    aTP_Lamp_aor: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aos____,type,
    aTP_Lamp_aos: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aot____,type,
    aTP_Lamp_aot: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aou____,type,
    aTP_Lamp_aou: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aov____,type,
    aTP_Lamp_aov: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aow____,type,
    aTP_Lamp_aow: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aox____,type,
    aTP_Lamp_aox: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoy____,type,
    aTP_Lamp_aoy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoz____,type,
    aTP_Lamp_aoz: 
      !>[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__apa____,type,
    aTP_Lamp_apa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apb____,type,
    aTP_Lamp_apb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apc____,type,
    aTP_Lamp_apc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apd____,type,
    aTP_Lamp_apd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ape____,type,
    aTP_Lamp_ape: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apf____,type,
    aTP_Lamp_apf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apg____,type,
    aTP_Lamp_apg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aph____,type,
    aTP_Lamp_aph: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__api____,type,
    aTP_Lamp_api: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apj____,type,
    aTP_Lamp_apj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apk____,type,
    aTP_Lamp_apk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apl____,type,
    aTP_Lamp_apl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apm____,type,
    aTP_Lamp_apm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apn____,type,
    aTP_Lamp_apn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apo____,type,
    aTP_Lamp_apo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__app____,type,
    aTP_Lamp_app: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apq____,type,
    aTP_Lamp_apq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apr____,type,
    aTP_Lamp_apr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aps____,type,
    aTP_Lamp_aps: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apt____,type,
    aTP_Lamp_apt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apu____,type,
    aTP_Lamp_apu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apv____,type,
    aTP_Lamp_apv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apw____,type,
    aTP_Lamp_apw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apx____,type,
    aTP_Lamp_apx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apy____,type,
    aTP_Lamp_apy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apz____,type,
    aTP_Lamp_apz: 
      !>[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__aqa____,type,
    aTP_Lamp_aqa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqb____,type,
    aTP_Lamp_aqb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqc____,type,
    aTP_Lamp_aqc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqd____,type,
    aTP_Lamp_aqd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqe____,type,
    aTP_Lamp_aqe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqf____,type,
    aTP_Lamp_aqf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqg____,type,
    aTP_Lamp_aqg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqh____,type,
    aTP_Lamp_aqh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqi____,type,
    aTP_Lamp_aqi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqj____,type,
    aTP_Lamp_aqj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqk____,type,
    aTP_Lamp_aqk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aql____,type,
    aTP_Lamp_aql: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqm____,type,
    aTP_Lamp_aqm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqn____,type,
    aTP_Lamp_aqn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqo____,type,
    aTP_Lamp_aqo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqp____,type,
    aTP_Lamp_aqp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aqq____,type,
    aTP_Lamp_aqq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aqr____,type,
    aTP_Lamp_aqr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqs____,type,
    aTP_Lamp_aqs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqt____,type,
    aTP_Lamp_aqt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqu____,type,
    aTP_Lamp_aqu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqv____,type,
    aTP_Lamp_aqv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqw____,type,
    aTP_Lamp_aqw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqx____,type,
    aTP_Lamp_aqx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqy____,type,
    aTP_Lamp_aqy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqz____,type,
    aTP_Lamp_aqz: 
      !>[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__ara____,type,
    aTP_Lamp_ara: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arb____,type,
    aTP_Lamp_arb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arc____,type,
    aTP_Lamp_arc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ard____,type,
    aTP_Lamp_ard: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__are____,type,
    aTP_Lamp_are: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arf____,type,
    aTP_Lamp_arf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arg____,type,
    aTP_Lamp_arg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arh____,type,
    aTP_Lamp_arh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ari____,type,
    aTP_Lamp_ari: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arj____,type,
    aTP_Lamp_arj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ark____,type,
    aTP_Lamp_ark: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arl____,type,
    aTP_Lamp_arl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__arm____,type,
    aTP_Lamp_arm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arn____,type,
    aTP_Lamp_arn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aro____,type,
    aTP_Lamp_aro: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arp____,type,
    aTP_Lamp_arp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arq____,type,
    aTP_Lamp_arq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arr____,type,
    aTP_Lamp_arr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ars____,type,
    aTP_Lamp_ars: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__art____,type,
    aTP_Lamp_art: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aru____,type,
    aTP_Lamp_aru: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__arv____,type,
    aTP_Lamp_arv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arw____,type,
    aTP_Lamp_arw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arx____,type,
    aTP_Lamp_arx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ary____,type,
    aTP_Lamp_ary: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__arz____,type,
    aTP_Lamp_arz: 
      !>[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__asa____,type,
    aTP_Lamp_asa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asb____,type,
    aTP_Lamp_asb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asc____,type,
    aTP_Lamp_asc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asd____,type,
    aTP_Lamp_asd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ase____,type,
    aTP_Lamp_ase: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asf____,type,
    aTP_Lamp_asf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asg____,type,
    aTP_Lamp_asg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ash____,type,
    aTP_Lamp_ash: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asi____,type,
    aTP_Lamp_asi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asj____,type,
    aTP_Lamp_asj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ask____,type,
    aTP_Lamp_ask: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asl____,type,
    aTP_Lamp_asl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asm____,type,
    aTP_Lamp_asm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asn____,type,
    aTP_Lamp_asn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aso____,type,
    aTP_Lamp_aso: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asp____,type,
    aTP_Lamp_asp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asq____,type,
    aTP_Lamp_asq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asr____,type,
    aTP_Lamp_asr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ass____,type,
    aTP_Lamp_ass: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ast____,type,
    aTP_Lamp_ast: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asu____,type,
    aTP_Lamp_asu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asv____,type,
    aTP_Lamp_asv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asw____,type,
    aTP_Lamp_asw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asx____,type,
    aTP_Lamp_asx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asy____,type,
    aTP_Lamp_asy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__asz____,type,
    aTP_Lamp_asz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__ata____,type,
    aTP_Lamp_ata: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atb____,type,
    aTP_Lamp_atb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atc____,type,
    aTP_Lamp_atc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atd____,type,
    aTP_Lamp_atd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ate____,type,
    aTP_Lamp_ate: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atf____,type,
    aTP_Lamp_atf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atg____,type,
    aTP_Lamp_atg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ath____,type,
    aTP_Lamp_ath: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ati____,type,
    aTP_Lamp_ati: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atj____,type,
    aTP_Lamp_atj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atk____,type,
    aTP_Lamp_atk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atl____,type,
    aTP_Lamp_atl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atm____,type,
    aTP_Lamp_atm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atn____,type,
    aTP_Lamp_atn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ato____,type,
    aTP_Lamp_ato: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atp____,type,
    aTP_Lamp_atp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atq____,type,
    aTP_Lamp_atq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atr____,type,
    aTP_Lamp_atr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ats____,type,
    aTP_Lamp_ats: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__att____,type,
    aTP_Lamp_att: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atu____,type,
    aTP_Lamp_atu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atv____,type,
    aTP_Lamp_atv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atw____,type,
    aTP_Lamp_atw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atx____,type,
    aTP_Lamp_atx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aty____,type,
    aTP_Lamp_aty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atz____,type,
    aTP_Lamp_atz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__aua____,type,
    aTP_Lamp_aua: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aub____,type,
    aTP_Lamp_aub: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auc____,type,
    aTP_Lamp_auc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aud____,type,
    aTP_Lamp_aud: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aue____,type,
    aTP_Lamp_aue: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auf____,type,
    aTP_Lamp_auf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aug____,type,
    aTP_Lamp_aug: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auh____,type,
    aTP_Lamp_auh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aui____,type,
    aTP_Lamp_aui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auj____,type,
    aTP_Lamp_auj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auk____,type,
    aTP_Lamp_auk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aul____,type,
    aTP_Lamp_aul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aum____,type,
    aTP_Lamp_aum: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aun____,type,
    aTP_Lamp_aun: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auo____,type,
    aTP_Lamp_auo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aup____,type,
    aTP_Lamp_aup: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__auq____,type,
    aTP_Lamp_auq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aur____,type,
    aTP_Lamp_aur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aus____,type,
    aTP_Lamp_aus: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aut____,type,
    aTP_Lamp_aut: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auu____,type,
    aTP_Lamp_auu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auv____,type,
    aTP_Lamp_auv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auw____,type,
    aTP_Lamp_auw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aux____,type,
    aTP_Lamp_aux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auy____,type,
    aTP_Lamp_auy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auz____,type,
    aTP_Lamp_auz: 
      !>[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__ava____,type,
    aTP_Lamp_ava: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avb____,type,
    aTP_Lamp_avb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avc____,type,
    aTP_Lamp_avc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avd____,type,
    aTP_Lamp_avd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ave____,type,
    aTP_Lamp_ave: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avf____,type,
    aTP_Lamp_avf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avg____,type,
    aTP_Lamp_avg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avh____,type,
    aTP_Lamp_avh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avi____,type,
    aTP_Lamp_avi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avj____,type,
    aTP_Lamp_avj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avk____,type,
    aTP_Lamp_avk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avl____,type,
    aTP_Lamp_avl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avm____,type,
    aTP_Lamp_avm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avn____,type,
    aTP_Lamp_avn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avo____,type,
    aTP_Lamp_avo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avp____,type,
    aTP_Lamp_avp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avq____,type,
    aTP_Lamp_avq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avr____,type,
    aTP_Lamp_avr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avs____,type,
    aTP_Lamp_avs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avt____,type,
    aTP_Lamp_avt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avu____,type,
    aTP_Lamp_avu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avv____,type,
    aTP_Lamp_avv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avw____,type,
    aTP_Lamp_avw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avx____,type,
    aTP_Lamp_avx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avy____,type,
    aTP_Lamp_avy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__avz____,type,
    aTP_Lamp_avz: 
      !>[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__awa____,type,
    aTP_Lamp_awa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awb____,type,
    aTP_Lamp_awb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awc____,type,
    aTP_Lamp_awc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awd____,type,
    aTP_Lamp_awd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awe____,type,
    aTP_Lamp_awe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awf____,type,
    aTP_Lamp_awf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awg____,type,
    aTP_Lamp_awg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awh____,type,
    aTP_Lamp_awh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awi____,type,
    aTP_Lamp_awi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awj____,type,
    aTP_Lamp_awj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awk____,type,
    aTP_Lamp_awk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awl____,type,
    aTP_Lamp_awl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awm____,type,
    aTP_Lamp_awm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awn____,type,
    aTP_Lamp_awn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awo____,type,
    aTP_Lamp_awo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awp____,type,
    aTP_Lamp_awp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awq____,type,
    aTP_Lamp_awq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awr____,type,
    aTP_Lamp_awr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aws____,type,
    aTP_Lamp_aws: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awt____,type,
    aTP_Lamp_awt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awu____,type,
    aTP_Lamp_awu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awv____,type,
    aTP_Lamp_awv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aww____,type,
    aTP_Lamp_aww: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awx____,type,
    aTP_Lamp_awx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__awy____,type,
    aTP_Lamp_awy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__awz____,type,
    aTP_Lamp_awz: 
      !>[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__axa____,type,
    aTP_Lamp_axa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axb____,type,
    aTP_Lamp_axb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axc____,type,
    aTP_Lamp_axc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axd____,type,
    aTP_Lamp_axd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axe____,type,
    aTP_Lamp_axe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axf____,type,
    aTP_Lamp_axf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axg____,type,
    aTP_Lamp_axg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axh____,type,
    aTP_Lamp_axh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axi____,type,
    aTP_Lamp_axi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axj____,type,
    aTP_Lamp_axj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axk____,type,
    aTP_Lamp_axk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axl____,type,
    aTP_Lamp_axl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axm____,type,
    aTP_Lamp_axm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axn____,type,
    aTP_Lamp_axn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axo____,type,
    aTP_Lamp_axo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axp____,type,
    aTP_Lamp_axp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axq____,type,
    aTP_Lamp_axq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axr____,type,
    aTP_Lamp_axr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axs____,type,
    aTP_Lamp_axs: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axt____,type,
    aTP_Lamp_axt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axu____,type,
    aTP_Lamp_axu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axv____,type,
    aTP_Lamp_axv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axw____,type,
    aTP_Lamp_axw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axx____,type,
    aTP_Lamp_axx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axy____,type,
    aTP_Lamp_axy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axz____,type,
    aTP_Lamp_axz: 
      !>[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__aya____,type,
    aTP_Lamp_aya: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayb____,type,
    aTP_Lamp_ayb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayc____,type,
    aTP_Lamp_ayc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayd____,type,
    aTP_Lamp_ayd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aye____,type,
    aTP_Lamp_aye: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayf____,type,
    aTP_Lamp_ayf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayg____,type,
    aTP_Lamp_ayg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ayh____,type,
    aTP_Lamp_ayh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ayi____,type,
    aTP_Lamp_ayi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayj____,type,
    aTP_Lamp_ayj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayk____,type,
    aTP_Lamp_ayk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayl____,type,
    aTP_Lamp_ayl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aym____,type,
    aTP_Lamp_aym: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayn____,type,
    aTP_Lamp_ayn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayo____,type,
    aTP_Lamp_ayo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayp____,type,
    aTP_Lamp_ayp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayq____,type,
    aTP_Lamp_ayq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayr____,type,
    aTP_Lamp_ayr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ays____,type,
    aTP_Lamp_ays: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayt____,type,
    aTP_Lamp_ayt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayu____,type,
    aTP_Lamp_ayu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ayv____,type,
    aTP_Lamp_ayv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayw____,type,
    aTP_Lamp_ayw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayx____,type,
    aTP_Lamp_ayx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ayy____,type,
    aTP_Lamp_ayy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayz____,type,
    aTP_Lamp_ayz: 
      !>[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__aza____,type,
    aTP_Lamp_aza: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azb____,type,
    aTP_Lamp_azb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azc____,type,
    aTP_Lamp_azc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azd____,type,
    aTP_Lamp_azd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aze____,type,
    aTP_Lamp_aze: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azf____,type,
    aTP_Lamp_azf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azg____,type,
    aTP_Lamp_azg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azh____,type,
    aTP_Lamp_azh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azi____,type,
    aTP_Lamp_azi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azj____,type,
    aTP_Lamp_azj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azk____,type,
    aTP_Lamp_azk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azl____,type,
    aTP_Lamp_azl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azm____,type,
    aTP_Lamp_azm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azn____,type,
    aTP_Lamp_azn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azo____,type,
    aTP_Lamp_azo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azp____,type,
    aTP_Lamp_azp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azq____,type,
    aTP_Lamp_azq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azr____,type,
    aTP_Lamp_azr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azs____,type,
    aTP_Lamp_azs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azt____,type,
    aTP_Lamp_azt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azu____,type,
    aTP_Lamp_azu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azv____,type,
    aTP_Lamp_azv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azw____,type,
    aTP_Lamp_azw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azx____,type,
    aTP_Lamp_azx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azy____,type,
    aTP_Lamp_azy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azz____,type,
    aTP_Lamp_azz: 
      !>[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__baa____,type,
    aTP_Lamp_baa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bab____,type,
    aTP_Lamp_bab: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bac____,type,
    aTP_Lamp_bac: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bad____,type,
    aTP_Lamp_bad: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bae____,type,
    aTP_Lamp_bae: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baf____,type,
    aTP_Lamp_baf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bag____,type,
    aTP_Lamp_bag: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bah____,type,
    aTP_Lamp_bah: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bai____,type,
    aTP_Lamp_bai: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baj____,type,
    aTP_Lamp_baj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bak____,type,
    aTP_Lamp_bak: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bal____,type,
    aTP_Lamp_bal: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bam____,type,
    aTP_Lamp_bam: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ban____,type,
    aTP_Lamp_ban: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bao____,type,
    aTP_Lamp_bao: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bap____,type,
    aTP_Lamp_bap: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baq____,type,
    aTP_Lamp_baq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bar____,type,
    aTP_Lamp_bar: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bas____,type,
    aTP_Lamp_bas: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bat____,type,
    aTP_Lamp_bat: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bau____,type,
    aTP_Lamp_bau: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bav____,type,
    aTP_Lamp_bav: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baw____,type,
    aTP_Lamp_baw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bax____,type,
    aTP_Lamp_bax: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bay____,type,
    aTP_Lamp_bay: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baz____,type,
    aTP_Lamp_baz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__bba____,type,
    aTP_Lamp_bba: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbb____,type,
    aTP_Lamp_bbb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbc____,type,
    aTP_Lamp_bbc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbd____,type,
    aTP_Lamp_bbd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbe____,type,
    aTP_Lamp_bbe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbf____,type,
    aTP_Lamp_bbf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__lm____,type,
    aTP_Lamp_lm: 
      !>[A: $tType,B: $tType] : fun(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] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lx____,type,
    aTP_Lamp_lx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ly____,type,
    aTP_Lamp_ly: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : ( 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] : fun(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] : fun(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] : ( 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] : ( 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] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mw____,type,
    aTP_Lamp_mw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mx____,type,
    aTP_Lamp_mx: 
      !>[A: $tType,B: $tType] : ( 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] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nf____,type,
    aTP_Lamp_nf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ng____,type,
    aTP_Lamp_ng: 
      !>[A: $tType,B: $tType] : fun(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] : ( 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] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ph____,type,
    aTP_Lamp_ph: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pv____,type,
    aTP_Lamp_pv: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__py____,type,
    aTP_Lamp_py: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qt____,type,
    aTP_Lamp_qt: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rb____,type,
    aTP_Lamp_rb: 
      !>[A: $tType,B: $tType] : fun(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] : ( 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] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ry____,type,
    aTP_Lamp_ry: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__te____,type,
    aTP_Lamp_te: 
      !>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ti____,type,
    aTP_Lamp_ti: 
      !>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uy____,type,
    aTP_Lamp_uy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uz____,type,
    aTP_Lamp_uz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__va____,type,
    aTP_Lamp_va: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ve____,type,
    aTP_Lamp_ve: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vf____,type,
    aTP_Lamp_vf: 
      !>[A: $tType,B: $tType] : fun(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] : ( 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] : ( 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] : fun(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] : fun(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] : fun(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] : fun(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] : fun(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] : ( 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] : ( 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_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
    bNF_Ca3754400796208372196lChain: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).

tff(sy_c_BNF__Def_OGr,type,
    bNF_Gr: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
    bNF_Greatest_Succ: 
      !>[A: $tType] : ( ( set(list(A)) * list(A) ) > set(A) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
    bNF_Greatest_image2: 
      !>[C: $tType,A: $tType,B: $tType] : ( ( set(C) * fun(C,A) * fun(C,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OFunc,type,
    bNF_Wellorder_Func: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(fun(A,B)) ) ).

tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
    basic_BNF_size_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * product_prod(A,B) ) > nat ) ).

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__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

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: $tType] : ( A > fun(nat,$o) ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
    bit_un1837492267222099188nd_num: fun(num,fun(num,option(num))) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
    bit_un6178654185764691216or_num: fun(num,fun(num,option(num))) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: fun(num,fun(num,option(num))) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: fun(num,fun(num,option(num))) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
    boolea2506097494486148201lgebra: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A ) > $o ) ).

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_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,$o) ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

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

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

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

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

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

tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
    complete_Inf_Inf: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
    complete_Sup_Sup: 
      !>[A: $tType] : fun(set(A),A) ).

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

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

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

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: ( real * real ) > complex ).

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

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

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

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
    condit941137186595557371_above: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
    condit1013018076250108175_below: 
      !>[A: $tType] : ( set(A) > $o ) ).

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_Oadjust__mod,type,
    adjust_mod: ( 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: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofiltercomap,type,
    filtercomap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) ) > filter(A) ) ).

tff(sy_c_Filter_Ofilterlim,type,
    filterlim: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofiltermap,type,
    filtermap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > filter(B) ) ).

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_Filter_Oprincipal,type,
    principal: 
      !>[A: $tType] : fun(set(A),filter(A)) ).

tff(sy_c_Filter_Oprod__filter,type,
    prod_filter: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * filter(B) ) > filter(product_prod(A,B)) ) ).

tff(sy_c_Finite__Set_Ocard,type,
    finite_card: 
      !>[B: $tType] : fun(set(B),nat) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
    finite6289374366891150609ommute: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

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

tff(sy_c_Finite__Set_Ofold,type,
    finite_fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).

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(fun(A,B),fun(A,C)) ) ).

tff(sy_c_Fun_Ofun__upd,type,
    fun_upd: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).

tff(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Omap__fun,type,
    map_fun: 
      !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(fun(A,B),fun(C,D)) ) ).

tff(sy_c_Fun_Ostrict__mono__on,type,
    strict_mono_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_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

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

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

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

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] : fun(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] : fun(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(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
    groups1962203154675924110t_prod: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > 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: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Int_OAbs__Integ,type,
    abs_Integ: fun(product_prod(nat,nat),int) ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: fun(int,product_prod(nat,nat)) ).

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: fun(int,nat) ).

tff(sy_c_Int_Opower__int,type,
    power_int: 
      !>[A: $tType] : ( ( A * int ) > A ) ).

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] : ( int > A ) ).

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

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

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

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_Obind,type,
    bind: 
      !>[A: $tType,B: $tType] : ( ( list(A) * fun(A,list(B)) ) > list(B) ) ).

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_Odistinct,type,
    distinct: 
      !>[A: $tType] : ( list(A) > $o ) ).

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

tff(sy_c_List_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

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

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)) > fun(nat,set(product_prod(list(A),list(A)))) ) ).

tff(sy_c_List_Olexord,type,
    lexord: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexordp,type,
    lexordp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olinorder__class_Oinsort__key,type,
    linorder_insort_key: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(B,fun(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 * list(A) ) > list(A) ) ).

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

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_Olistrel1,type,
    listrel1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olistrel1p,type,
    listrel1p: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > $o ) ).

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_Oremdups,type,
    remdups: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremdups__adj,type,
    remdups_adj: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_OremoveAll,type,
    removeAll: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

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

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( list(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_Oshuffles,type,
    shuffles: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).

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

tff(sy_c_List_Ounion,type,
    union: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

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),$o)) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Ocompow,type,
    compow: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Nat_Ofunpow,type,
    funpow: 
      !>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).

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

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

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,$o) ) ).

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_Olist__encode,type,
    nat_list_encode: list(nat) > nat ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),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_Oneg__numeral__class_Osub,type,
    neg_numeral_sub: 
      !>[A: $tType] : ( ( num * num ) > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun(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_Num_Oring__1__class_Oiszero,type,
    ring_1_iszero: 
      !>[A: $tType] : ( A > $o ) ).

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

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_Omap__option,type,
    map_option: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(option(A),option(Aa)) ) ).

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

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

tff(sy_c_Option_Othese,type,
    these: 
      !>[A: $tType] : ( set(option(A)) > set(A) ) ).

tff(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
    order_532582986084564980_cclfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

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

tff(sy_c_Orderings_Oord_OLeast,type,
    least: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,$o) ) > A ) ).

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

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

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

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

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
    order_strict_mono: 
      !>[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_OSigma,type,
    product_Sigma: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Product__Type_Oapfst,type,
    product_apfst: 
      !>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).

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

tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,$o) ) ).

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

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

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

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

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

tff(sy_c_Real__Vector__Spaces_OReals,type,
    real_Vector_Reals: 
      !>[A: $tType] : set(A) ).

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_Obounded__linear__axioms,type,
    real_V4916620083959148203axioms: 
      !>[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_Relation_OId__on,type,
    id_on: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_Orelcomp,type,
    relcomp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).

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: $tType] : fun(A,fun(A,$o)) ).

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: $tType] : fun($o,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: $tType] : ( fun(nat,A) > fun(A,$o) ) ).

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

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

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > 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 > fun(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] : fun(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] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : fun(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] : fun(A,set(A)) ).

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

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o ) > fun($o,char) ).

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

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_Ogenerate__topology,type,
    topolo8378437560675496660pology: 
      !>[A: $tType] : ( set(set(A)) > fun(set(A),$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: $tType] : fun(set(A),$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_Oclosed,type,
    topolo7761053866217962861closed: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type,
    topolo2193935891317330818ompact: 
      !>[A: $tType] : ( set(A) > $o ) ).

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_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
    topolo6688025880775521714ounded: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type,
    topolo7806501430040627800ormity: 
      !>[A: $tType] : filter(product_prod(A,A)) ).

tff(sy_c_Topological__Spaces_Ouniformly__continuous__on,type,
    topolo6026614971017936543ous_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > $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_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortrancl,type,
    transitive_rtrancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

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),$o)) ).

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),$o)) ).

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,$o)) ).

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),$o)) ).

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),$o)) ).

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,$o)) ).

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,$o)) ).

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__rel,type,
    vEBT_T_p_r_e_d_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

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),$o)) ).

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),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > 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: vEBT_VEBT > fun(nat,$o) ).

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),$o)) ).

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),$o)) ).

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),$o)) ).

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,$o)) ).

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,$o)) ).

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),$o)) ).

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 > $o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

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,$o) ).

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),$o)) ).

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) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option(nat) * option(nat) ) > $o ).

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,$o)) ).

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,$o)) ).

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),$o)) ).

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),$o)) ).

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

tff(sy_c_Wellfounded_Olex__prod,type,
    lex_prod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

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

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wfrec_Osame__fst,type,
    same_fst: 
      !>[A: $tType,B: $tType] : ( ( fun(A,$o) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

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

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : ( A > fun(A,$o) ) ).

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

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list(vEBT_VEBT) ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (9566)
tff(fact_0_False,axiom,
    ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),ma),xa) ).

% False
tff(fact_1_bit__split__inv,axiom,
    ! [Xb: nat,D2: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xb,D2),vEBT_VEBT_low(Xb,D2),D2) = Xb ).

% bit_split_inv
tff(fact_2_True,axiom,
    ( ( vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) != none(nat) )
    & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% True
tff(fact_3__C4_Ohyps_C_I7_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),mi),ma) ).

% "4.hyps"(7)
tff(fact_4__C4_Ohyps_C_I8_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),deg)) ).

% "4.hyps"(8)
tff(fact_5_add__self__div__2,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Mb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Mb ).

% add_self_div_2
tff(fact_6_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_7_semiring__norm_I86_J,axiom,
    ! [Mb: num] : aa(num,num,bit1,Mb) != one2 ).

% semiring_norm(86)
tff(fact_8_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one2 != aa(num,num,bit1,Nb) ).

% semiring_norm(84)
tff(fact_9_semiring__norm_I89_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,bit1,Mb) != aa(num,num,bit0,Nb) ).

% semiring_norm(89)
tff(fact_10_semiring__norm_I88_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,bit0,Mb) != aa(num,num,bit1,Nb) ).

% semiring_norm(88)
tff(fact_11_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),Nb) ) ).

% numeral_Bit1_div_2
tff(fact_12_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = Xb ) ).

% field_sum_of_halves
tff(fact_13_semiring__norm_I85_J,axiom,
    ! [Mb: num] : aa(num,num,bit0,Mb) != one2 ).

% semiring_norm(85)
tff(fact_14_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one2 != aa(num,num,bit0,Nb) ).

% semiring_norm(83)
tff(fact_15_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ) ).

% numeral_plus_numeral
tff(fact_16_greater__shift,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xb)
    <=> vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Xb),aa(nat,option(nat),some(nat),Y)) ) ).

% greater_shift
tff(fact_17_max__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat] :
      ( vEBT_VEBT_max_in_set(Xs,Xb)
    <=> ( aa(set(nat),$o,member(nat,Xb),Xs)
        & ! [X: nat] :
            ( aa(set(nat),$o,member(nat,X),Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Xb) ) ) ) ).

% max_in_set_def
tff(fact_18_min__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat] :
      ( vEBT_VEBT_min_in_set(Xs,Xb)
    <=> ( aa(set(nat),$o,member(nat,Xb),Xs)
        & ! [X: nat] :
            ( aa(set(nat),$o,member(nat,X),Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),X) ) ) ) ).

% min_in_set_def
tff(fact_19_power__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xb),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_power,aa(nat,option(nat),some(nat),Xb)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% power_shift
tff(fact_20_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Mb: num,Nb: num] :
          ( ( aa(num,A,numeral_numeral(A),Mb) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( Mb = Nb ) ) ) ).

% numeral_eq_iff
tff(fact_21_semiring__norm_I87_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa(num,num,bit0,Mb) = aa(num,num,bit0,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(87)
tff(fact_22_semiring__norm_I90_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa(num,num,bit1,Mb) = aa(num,num,bit1,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(90)
tff(fact_23__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),deg) ).

% \<open>2 \<le> deg\<close>
tff(fact_24_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),aa(num,num,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),aa(num,num,bit0,one2))),A2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2) ).

% pow_sum
tff(fact_25_high__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_high(Xb,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% high_def
tff(fact_26_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Maa,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb)) ) ).

% high_bound_aux
tff(fact_27_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb) ) ) ).

% numeral_le_iff
tff(fact_28_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ) ).

% numeral_less_iff
tff(fact_29_semiring__norm_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,Mb)),aa(num,num,bit0,Nb)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ).

% semiring_norm(6)
tff(fact_30_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( one2 = Nb ) ) ) ).

% one_eq_numeral_iff
tff(fact_31_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(num,A,numeral_numeral(A),Nb) = one_one(A) )
        <=> ( Nb = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_32_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_33_semiring__norm_I2_J,axiom,
    aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = aa(num,num,bit0,one2) ).

% semiring_norm(2)
tff(fact_34_semiring__norm_I7_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,Mb)),aa(num,num,bit1,Nb)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ).

% semiring_norm(7)
tff(fact_35_semiring__norm_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,Mb)),aa(num,num,bit0,Nb)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ).

% semiring_norm(9)
tff(fact_36_pred__max,axiom,
    ! [Dega: nat,Maa: nat,Xb: nat,Mia: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb)
       => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = aa(nat,option(nat),some(nat),Maa) ) ) ) ).

% pred_max
tff(fact_37_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),one2) ) ) ).

% numeral_le_one_iff
tff(fact_38_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),Nb) ) ) ).

% one_less_numeral_iff
tff(fact_39_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit0,Nb)) = aa(num,num,bit1,Nb) ).

% semiring_norm(3)
tff(fact_40_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,Nb)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ).

% semiring_norm(4)
tff(fact_41_semiring__norm_I5_J,axiom,
    ! [Mb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,Mb)),one2) = aa(num,num,bit1,Mb) ).

% semiring_norm(5)
tff(fact_42_semiring__norm_I8_J,axiom,
    ! [Mb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,Mb)),one2) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),one2)) ).

% semiring_norm(8)
tff(fact_43_semiring__norm_I10_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,Mb)),aa(num,num,bit1,Nb)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)),one2)) ).

% semiring_norm(10)
tff(fact_44_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,A2),aa(fun(A,$o),set(A),collect(A),P))
    <=> aa(A,$o,P,A2) ) ).

% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)) = A3 ).

% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
        <=> aa(A,$o,Q,X2) )
     => ( aa(fun(A,$o),set(A),collect(A),P) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_47_ext,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] :
      ( ! [X2: A] : aa(A,B,F2,X2) = aa(A,B,G,X2)
     => ( F2 = G ) ) ).

% ext
tff(fact_48_lesseq__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Y)
    <=> vEBT_VEBT_lesseq(aa(nat,option(nat),some(nat),Xb),aa(nat,option(nat),some(nat),Y)) ) ).

% lesseq_shift
tff(fact_49_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),aa(num,num,bit0,one2)) ) ) ).

% one_add_one
tff(fact_50_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb)) ) ).

% one_plus_numeral
tff(fact_51_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ) ).

% numeral_plus_one
tff(fact_52_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_53_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),one_one(A)) ) ).

% less_numeral_extra(4)
tff(fact_54_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% one_le_numeral
tff(fact_55_add__One__commute,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb) = aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2) ).

% add_One_commute
tff(fact_56_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_57_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),K)) ) ).

% div_le_mono
tff(fact_58_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_59_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_60_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_61_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_62_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_63_mint__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_t(Ta)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% mint_bound
tff(fact_64_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)) = 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),Nb)),aa(num,A,numeral_numeral(A),Nb))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_65_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),aa(num,num,bit0,one2)) ).

% nat_1_add_1
tff(fact_66_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ).

% field_less_half_sum
tff(fact_67_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Nb)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% numeral_Bit0
tff(fact_68_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_69_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X22: num] : Y != aa(num,num,bit0,X22)
       => ~ ! [X3: num] : Y != aa(num,num,bit1,X3) ) ) ).

% num.exhaust
tff(fact_70_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),Nb) ) ).

% numeral_Bit0_div_2
tff(fact_71_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
       => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = 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),Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_72_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Y) ) ) ) ).

% power_increasing_iff
tff(fact_73_succ__min,axiom,
    ! [Dega: nat,Xb: nat,Mia: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia)
       => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = aa(nat,option(nat),some(nat),Mia) ) ) ) ).

% succ_min
tff(fact_74_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_75__C1_C,axiom,
    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)),deg,treeList,summary),xa) = 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,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),
      $ite(
        ( ( vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
        $let(
          pr: option(nat),
          pr:= vEBT_vebt_pred(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,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(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(nat))),
            $ite(pr = 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),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),aa(option(nat),nat,the2(nat),pr)))))) ) )) ).

% "1"
tff(fact_76_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),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)))) ) ) ) ).

% ex_power_ivl2
tff(fact_77_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),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)))) ) ) ) ).

% ex_power_ivl1
tff(fact_78_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_79_maxt__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_a_x_t(Ta)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% maxt_bound
tff(fact_80_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) )
          <=> ( Mb = Nb ) ) ) ) ).

% power_inject_exp
tff(fact_81_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Dega)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_82_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Mb: nat,Nb: 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),aa(num,num,bit0,one2))),Mb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = 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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ).

% div_exp_eq
tff(fact_83_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_84_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),Nb) = one_one(A) ) ).

% power_one
tff(fact_85_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_86_semiring__norm_I78_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit0,Mb)),aa(num,num,bit0,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ).

% semiring_norm(78)
tff(fact_87_semiring__norm_I71_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit0,Mb)),aa(num,num,bit0,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb) ) ).

% semiring_norm(71)
tff(fact_88_semiring__norm_I75_J,axiom,
    ! [Mb: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),one2) ).

% semiring_norm(75)
tff(fact_89_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),one2),Nb) ).

% semiring_norm(68)
tff(fact_90_semiring__norm_I80_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit1,Mb)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ).

% semiring_norm(80)
tff(fact_91_semiring__norm_I73_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,Mb)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb) ) ).

% semiring_norm(73)
tff(fact_92_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),aa(num,num,bit0,Nb)) ).

% semiring_norm(76)
tff(fact_93_semiring__norm_I69_J,axiom,
    ! [Mb: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit0,Mb)),one2) ).

% semiring_norm(69)
tff(fact_94_semiring__norm_I81_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit1,Mb)),aa(num,num,bit0,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ).

% semiring_norm(81)
tff(fact_95_semiring__norm_I72_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit0,Mb)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb) ) ).

% semiring_norm(72)
tff(fact_96_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),aa(num,num,bit1,Nb)) ).

% semiring_norm(77)
tff(fact_97_semiring__norm_I70_J,axiom,
    ! [Mb: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,Mb)),one2) ).

% semiring_norm(70)
tff(fact_98__C4_Ohyps_C_I5_J,axiom,
    ! [I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m))
     => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I)),X_1)
      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(summary),I) ) ) ).

% "4.hyps"(5)
tff(fact_99_semiring__norm_I79_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),aa(num,num,bit0,Mb)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb) ) ).

% semiring_norm(79)
tff(fact_100_semiring__norm_I74_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,Mb)),aa(num,num,bit0,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ).

% semiring_norm(74)
tff(fact_101__C4_Ohyps_C_I6_J,axiom,
    ( ( mi = ma )
   => ! [X4: vEBT_VEBT] :
        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) ) ).

% "4.hyps"(6)
tff(fact_102_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_103_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Nb)) ).

% two_realpow_ge_one
tff(fact_104_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Xb),one2)
    <=> ( Xb = one2 ) ) ).

% le_num_One_iff
tff(fact_105_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] :
          aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)) = $let(
            m2: A,
            m2:= aa(num,A,numeral_numeral(A),Nb),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),m2),m2) ) ) ).

% numeral_code(2)
tff(fact_106_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] :
          aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)) = $let(
            m2: A,
            m2:= aa(num,A,numeral_numeral(A),Nb),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m2),m2)),one_one(A)) ) ) ).

% numeral_code(3)
tff(fact_107_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,Nb: 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)),Nb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ).

% power_divide
tff(fact_108_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% one_le_power
tff(fact_109_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: 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)),Nb) = 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),Nb)) ) ).

% power_one_over
tff(fact_110_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N2)) ) ) ) ).

% power_strict_increasing
tff(fact_111_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% power_less_imp_less_exp
tff(fact_112_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N2)) ) ) ) ).

% power_increasing
tff(fact_113_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% power_le_imp_le_exp
tff(fact_114_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% one_power2
tff(fact_115_less__exp,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% less_exp
tff(fact_116_self__le__ge2__pow,axiom,
    ! [K: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Mb)) ) ).

% self_le_ge2_pow
tff(fact_117_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_118_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_119__C0_C,axiom,
    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)),deg,treeList,summary),xa) = 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)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2)))))),one_one(nat))),
      $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList)),
        $let(
          minlow: option(nat),
          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
            $ite(
              ( ( minlow != none(nat) )
              & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),minlow) ),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
              $let(
                pr: option(nat),
                pr:= vEBT_vebt_pred(summary,vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,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(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(nat))),
                  $ite(pr = 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),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
        one_one(nat) )) ).

% "0"
tff(fact_120_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list(vEBT_VEBT),Xb: nat] :
      ( vEBT_V5917875025757280293ildren(Nb,TreeLista,Xb)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,Nb))),vEBT_VEBT_low(Xb,Nb)) ) ).

% in_children_def
tff(fact_121__C4_Ohyps_C_I9_J,axiom,
    ( ( mi != ma )
   => ! [I: nat] :
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m))
       => ( ( ( vEBT_VEBT_high(ma,na) = I )
           => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I)),vEBT_VEBT_low(ma,na)) )
          & ! [X4: nat] :
              ( ( ( vEBT_VEBT_high(X4,na) = I )
                & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I)),vEBT_VEBT_low(X4,na)) )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),mi),X4)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),ma) ) ) ) ) ) ).

% "4.hyps"(9)
tff(fact_122_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Dega)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => aa(nat,$o,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),Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_123_pred__list__to__short,axiom,
    ! [Dega: nat,Xb: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_124_succ__list__to__short,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list(vEBT_VEBT),Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Xb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_125_option_Ocollapse,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) = Option ) ) ).

% option.collapse
tff(fact_126_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_127_bit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : vEBT_VEBT_bit_concat(H,L,D2) = 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),aa(num,num,bit0,one2))),D2))),L) ).

% bit_concat_def
tff(fact_128_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
     => ( 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),aa(num,num,bit0,one2))),Nb))),Xb),Nb) = Xb ) ) ).

% low_inv
tff(fact_129_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
     => ( 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),aa(num,num,bit0,one2))),Nb))),Xb),Nb) = Y ) ) ).

% high_inv
tff(fact_130_vebt__mint_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Mia) ).

% vebt_mint.simps(3)
tff(fact_131_deg__deg__n,axiom,
    ! [Info: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya),Nb)
     => ( Dega = Nb ) ) ).

% deg_deg_n
tff(fact_132__C4_Ohyps_C_I3_J,axiom,
    m = na ).

% "4.hyps"(3)
tff(fact_133__C4_Ohyps_C_I1_J,axiom,
    vEBT_invar_vebt(summary,m) ).

% "4.hyps"(1)
tff(fact_134__C4_Ohyps_C_I4_J,axiom,
    deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) ).

% "4.hyps"(4)
tff(fact_135_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Nb: nat] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% inthall
tff(fact_136__C4_Ohyps_C_I2_J,axiom,
    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),aa(num,num,bit0,one2))),m) ).

% "4.hyps"(2)
tff(fact_137_option_Oinject,axiom,
    ! [A: $tType,X23: A,Y2: A] :
      ( ( aa(A,option(A),some(A),X23) = aa(A,option(A),some(A),Y2) )
    <=> ( X23 = Y2 ) ) ).

% option.inject
tff(fact_138_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),Dega,TreeLista,Summarya),Dega)
     => ( ( Mia = Maa )
       => ( ! [X4: vEBT_VEBT] :
              ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
             => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) )
          & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_139_geqmaxNone,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat,Xb: 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),Mia),Maa)),Dega,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Maa),Xb)
       => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_140_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb)) ) ).

% numeral_times_numeral
tff(fact_141_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_142_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( aa(nat,option(nat),some(nat),Mb) = vEBT_vebt_mint(Ta) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ) ).

% misiz
tff(fact_143_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ) ).

% helpyd
tff(fact_144_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ) ).

% helpypredd
tff(fact_145_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
         => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_146_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_147_not__Some__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ! [Y3: A] : Xb != aa(A,option(A),some(A),Y3)
    <=> ( Xb = none(A) ) ) ).

% not_Some_eq
tff(fact_148_not__None__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ( Xb != none(A) )
    <=> ? [Y3: A] : Xb = aa(A,option(A),some(A),Y3) ) ).

% not_None_eq
tff(fact_149_set__n__deg__not__0,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Mb: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_150_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: 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),Mia),Maa)),Dega,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_151_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_152_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_153_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),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_154_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_155_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),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_156_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_157_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Mb: num,Nb: 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),Mb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),Nb))) = 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),Mb),Nb))) ) ).

% power_add_numeral
tff(fact_158_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Mb: num,Nb: 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),Mb))),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),Nb))),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),Mb),Nb)))),B2) ) ).

% power_add_numeral2
tff(fact_159__C4_OIH_C_I1_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
     => ( vEBT_invar_vebt(X4,na)
        & ! [Xa: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d(X4,Xa)),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,X4))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2))))))) ) ) ).

% "4.IH"(1)
tff(fact_160_complete__real,axiom,
    ! [S: set(real)] :
      ( ? [X4: real] : aa(set(real),$o,member(real,X4),S)
     => ( ? [Z2: real] :
          ! [X2: real] :
            ( aa(set(real),$o,member(real,X2),S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Z2) )
       => ? [Y4: real] :
            ( ! [X4: real] :
                ( aa(set(real),$o,member(real,X4),S)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),Y4) )
            & ! [Z2: real] :
                ( ! [X2: real] :
                    ( aa(set(real),$o,member(real,X2),S)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Z2) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),Z2) ) ) ) ) ).

% complete_real
tff(fact_161_real__arch__pow,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N)) ) ).

% real_arch_pow
tff(fact_162_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_163_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A2: A,B2: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,F2),aa(A,option(A),some(A),A2)),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F2,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_164_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),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),Nb)) ) ).

% power_commutes
tff(fact_165_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,Nb: 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)),Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ).

% power_mult_distrib
tff(fact_166_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) ) ) ) ).

% power_commuting_commutes
tff(fact_167_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_168_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Mb: nat,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb)),Nb) ) ).

% power_mult
tff(fact_169_left__add__mult__distrib,axiom,
    ! [I2: 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),I2),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),I2),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_170_div__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Q2) ).

% div_mult2_eq
tff(fact_171_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_172_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_173_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),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),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_174_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Mb: nat,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_add
tff(fact_175_less__mult__imp__div__less,axiom,
    ! [Mb: nat,I2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),I2) ) ).

% less_mult_imp_div_less
tff(fact_176_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),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),Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_177_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_178_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_179_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,A)),Xaa: option(A),Xba: option(A),Y: option(A)] :
      ( ( aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Xb),Xaa),Xba) = Y )
     => ( ( ( Xaa = none(A) )
         => ( Y != none(A) ) )
       => ( ( ? [V2: A] : Xaa = aa(A,option(A),some(A),V2)
           => ( ( Xba = none(A) )
             => ( Y != none(A) ) ) )
         => ~ ! [A4: A] :
                ( ( Xaa = aa(A,option(A),some(A),A4) )
               => ! [B3: A] :
                    ( ( Xba = aa(A,option(A),some(A),B3) )
                   => ( Y != aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Xb,A4),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_180_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_181_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),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),Nb))) ) ) ).

% power_gt1_lemma
tff(fact_182_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))) ) ) ).

% power_less_power_Suc
tff(fact_183_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_184_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),aa(num,num,bit0,W))) = $let(
            w: A,
            w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W)),
            aa(A,A,aa(A,fun(A,A),times_times(A),w),w) ) ) ).

% power_numeral_even
tff(fact_185_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))) = $let(
            w: A,
            w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W)),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),w)),w) ) ) ).

% power_numeral_odd
tff(fact_186_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_187_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_188_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),aa(num,num,bit0,one2))),A2)),B2) ) ).

% left_add_twice
tff(fact_189_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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2_right
tff(fact_190_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),aa(num,num,bit0,one2))),Z) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2
tff(fact_191_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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% power2_eq_square
tff(fact_192_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,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),Xb),Xb)),Xb)),Xb) ) ).

% power4_eq_xxxx
tff(fact_193_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: 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),aa(num,num,bit0,one2))),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power_even_eq
tff(fact_194_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_195_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: 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),Xb),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),Xb)),Y)) ) ).

% power2_sum
tff(fact_196_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,Xb: option(A),P: fun(option(A),fun(option(B),$o)),Y: option(B)] :
      ( ( ( Xb = none(A) )
       => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) )
     => ( ( ( Y = none(B) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) )
       => ( ! [A4: A,B3: B] :
              ( ( Xb = aa(A,option(A),some(A),A4) )
             => ( ( Y = aa(B,option(B),some(B),B3) )
               => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) ) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_197_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ! [X_1: option(A)] : aa(option(A),$o,P,X_1)
    <=> ( aa(option(A),$o,P,none(A))
        & ! [X: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X)) ) ) ).

% split_option_all
tff(fact_198_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ? [X_1: option(A)] : aa(option(A),$o,P,X_1)
    <=> ( aa(option(A),$o,P,none(A))
        | ? [X: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X)) ) ) ).

% split_option_ex
tff(fact_199_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_200_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_201_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X23: A] : none(A) != aa(A,option(A),some(A),X23) ).

% option.distinct(1)
tff(fact_202_option_Osel,axiom,
    ! [A: $tType,X23: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X23)) = X23 ).

% option.sel
tff(fact_203_option_Oexpand,axiom,
    ! [A: $tType,Option: option(A),Option2: option(A)] :
      ( ( ( Option = none(A) )
      <=> ( Option2 = none(A) ) )
     => ( ( ( Option != none(A) )
         => ( ( Option2 != none(A) )
           => ( aa(option(A),A,the2(A),Option) = aa(option(A),A,the2(A),Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_204_option_Oexhaust__sel,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) ) ) ).

% option.exhaust_sel
tff(fact_205_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Xb) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta),Xb) ) ) ).

% mint_corr
tff(fact_206_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta),Xb)
       => ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Xb) ) ) ) ).

% mint_sound
tff(fact_207_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( vEBT_invar_vebt(Summarya,Mb)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb))
                   => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),X_1)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( vEBT_VEBT_high(X2,Nb) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(X2,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),X2)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Maa) ) ) ) ) )
                       => 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),Mia),Maa)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_208_nested__mint,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: 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),Mia),Maa)),Dega,TreeLista,Summarya),Nb)
     => ( ( Nb = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Mia)
         => ( ( Maa != Mia )
           => aa(nat,$o,aa(nat,fun(nat,$o),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),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_209_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Maa)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ) ).

% member_inv
tff(fact_210_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),Dega,TreeLista,Summarya),Dega)
     => ( ( Mia != Maa )
       => ( aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Summarya)) = vEBT_VEBT_high(Maa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% summaxma
tff(fact_211__C4_OIH_C_I2_J,axiom,
    ! [Xb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d(summary,Xb)),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,summary))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2))))))) ).

% "4.IH"(2)
tff(fact_212_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( vEBT_invar_vebt(Summarya,Mb)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb) )
             => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X2: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_213_mintlistlength,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: 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),Mia),Maa)),Dega,TreeLista,Summarya),Nb)
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Maa)
          & ? [M: nat] :
              ( ( aa(nat,option(nat),some(nat),M) = vEBT_vebt_mint(Summarya) )
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% mintlistlength
tff(fact_214_two__powr__height__bound__deg,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ).

% two_powr_height_bound_deg
tff(fact_215_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),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),aa(num,num,bit0,one2))),Xb)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% sum_squares_bound
tff(fact_216_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xb,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),aa(num,num,bit0,one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xb,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),aa(num,num,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(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),one_one(nat))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                    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,bit0,one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                      $ite(
                        ( ( minlow != none(nat) )
                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                        $let(
                          pr: option(nat),
                          pr:= vEBT_vebt_pred(Summarya,h),
                          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(Summarya,h))),one_one(nat))),
                            $ite(pr = 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),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_217_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)
     => ~ ! [N: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) ) ).

% even_odd_cases
tff(fact_218_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_219_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))
     => ? [Info2: option(product_prod(nat,nat)),TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,Nb)),TreeList,S2) ) ).

% deg_SUcn_Node
tff(fact_220_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_221_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_222_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xb) )
     => aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb) ) ).

% maxbmo
tff(fact_223_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(Ta),Maxi) ) ) ).

% mint_member
tff(fact_224_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(Ta),Maxi) ) ) ).

% maxt_member
tff(fact_225_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_226_height__compose__summary,axiom,
    ! [Summarya: vEBT_VEBT,Info: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),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,Summarya))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Dega,TreeLista,Summarya))) ).

% height_compose_summary
tff(fact_227_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Mini) )
       => ( aa(nat,$o,vEBT_vebt_member(Ta),Xb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mini),Xb) ) ) ) ).

% mint_corr_help
tff(fact_228_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Maxi) )
       => ( aa(nat,$o,vEBT_vebt_member(Ta),Xb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_229_height__compose__child,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list(vEBT_VEBT),Info: option(product_prod(nat,nat)),Dega: nat,Summarya: vEBT_VEBT] :
      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,Ta),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
     => aa(nat,$o,aa(nat,fun(nat,$o),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,Ta))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Dega,TreeLista,Summarya))) ) ).

% height_compose_child
tff(fact_230_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta),Xb)
       => ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xb) ) ) ) ).

% maxt_sound
tff(fact_231_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xb) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta),Xb) ) ) ).

% maxt_corr
tff(fact_232_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(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),aa(num,num,bit0,one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)) ) ) ).

% power_minus_is_div
tff(fact_233_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
     => ( vEBT_invar_vebt(Tree,Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ) ).

% member_bound
tff(fact_234_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
         => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(Ta,Xb)),Y)
           => ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_235_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Ta),Xb)
      <=> aa(set(nat),$o,member(nat,Xb),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_236_semiring__norm_I13_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,Mb)),aa(num,num,bit0,Nb)) = aa(num,num,bit0,aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb))) ).

% semiring_norm(13)
tff(fact_237_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),Nb) = Nb ).

% semiring_norm(12)
tff(fact_238_semiring__norm_I11_J,axiom,
    ! [Mb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),Mb),one2) = Mb ).

% semiring_norm(11)
tff(fact_239_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_240_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,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(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_241_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,aa(A,fun(A,A),minus_minus(A),A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(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_242_num__double,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,one2)),Nb) = aa(num,num,bit0,Nb) ).

% num_double
tff(fact_243_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Mb: num,Nb: 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),Mb))),aa(num,nat,numeral_numeral(nat),Nb)) = 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),Mb),Nb))) ) ).

% power_mult_numeral
tff(fact_244_semiring__norm_I15_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,Mb)),aa(num,num,bit0,Nb)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,Mb)),Nb)) ).

% semiring_norm(15)
tff(fact_245_semiring__norm_I14_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,Mb)),aa(num,num,bit1,Nb)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),Mb),aa(num,num,bit1,Nb))) ).

% semiring_norm(14)
tff(fact_246_Suc__numeral,axiom,
    ! [Nb: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ).

% Suc_numeral
tff(fact_247_semiring__norm_I16_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,Mb)),aa(num,num,bit1,Nb)) = 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),Mb),Nb)),aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb)))) ).

% semiring_norm(16)
tff(fact_248_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) = aa(nat,nat,suc,aa(nat,nat,suc,Nb)) ).

% add_2_eq_Suc
tff(fact_249_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_250_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% Suc_1
tff(fact_251_div2__Suc__Suc,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Mb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% div2_Suc_Suc
tff(fact_252_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
        & ! [Z3: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Ta),Z3)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Z3) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Z3) ) ) ) ).

% succ_member
tff(fact_253_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xb)
        & ! [Z3: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Ta),Z3)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z3),Xb) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Z3),Y) ) ) ) ).

% pred_member
tff(fact_254_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Sx) ) ) ).

% succ_corr
tff(fact_255_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Px: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Px) ) ) ).

% pred_corr
tff(fact_256_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_257_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: 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,Mb)))),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))),Mb)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_258_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D2: A] : aa(A,A,aa(A,fun(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),D2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2)) ) ).

% add_diff_add
tff(fact_259_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Xb: A,Y: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),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),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_260_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_261_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_Suc
tff(fact_262_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),A2) ) ).

% power_Suc2
tff(fact_263_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_264_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_265_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))) ) ) ).

% power_gt1
tff(fact_266_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Nb)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_267_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,nat,aa(nat,fun(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),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),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),minus_minus(nat),J),I2)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_268_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2)
     => ( aa(nat,nat,aa(nat,fun(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),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb)) = aa(nat,nat,aa(nat,fun(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,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_269_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),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),minus_minus(nat),J),I2)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_270_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),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,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_271_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb) )
      <=> ( Mb = 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),minus_minus(nat),J),I2)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_272_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb) )
      <=> ( 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),minus_minus(nat),I2),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_273_four__x__squared,axiom,
    ! [Xb: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% four_x_squared
tff(fact_274_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_commute
tff(fact_275_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),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),minus_minus(nat),J),I2)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_276_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),I2),U)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),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,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_277_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Nb) ).

% Suc3_eq_add_3
tff(fact_278_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q2)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_279_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = 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))),Nb) ).

% Suc_nat_number_of_add
tff(fact_280_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,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),D2))),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),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_281_diff__le__diff__pow,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_282_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Maa) ).

% vebt_maxt.simps(3)
tff(fact_283_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb)))),Nb) = 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))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_284_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: 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),aa(num,num,bit0,one2))),Nb))) = 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),Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% power_odd_eq
tff(fact_285_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),Xb)),Y)) ) ).

% power2_diff
tff(fact_286_height__node,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: 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),Mia),Maa)),Dega,TreeLista,Summarya),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),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),Mia),Maa)),Dega,TreeLista,Summarya))) ) ).

% height_node
tff(fact_287_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( vEBT_invar_vebt(Summarya,Mb)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb) )
         => ( ( Mb = aa(nat,nat,suc,Nb) )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb) )
             => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X2: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_288_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( vEBT_invar_vebt(Summarya,Mb)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb) )
         => ( ( Mb = aa(nat,nat,suc,Nb) )
           => ( ( Dega = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb))
                   => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),X_1)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( vEBT_VEBT_high(X2,Nb) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(X2,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),X2)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Maa) ) ) ) ) )
                       => 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),Mia),Maa)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_289_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
              $false,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,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),aa(num,num,bit0,one2)))),
                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_VEBT_low(Xb,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),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_290_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))),I2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_291_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_292_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,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
        $ite(
          Xb = Mia,
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
            $ite(
              Xb = Maa,
              one_one(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
                  one_one(nat),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                    $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
                      one_one(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,bit0,aa(num,num,bit0,one2))))),
                        $let(
                          h: nat,
                          h:= vEBT_VEBT_high(Xb,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),aa(num,num,bit0,one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),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,TreeLista),h),vEBT_VEBT_low(Xb,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),aa(num,num,bit0,one2)))))),one_one(nat)) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_293_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_294_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_295_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_296_diff__Suc__1,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Nb)),one_one(nat)) = Nb ).

% diff_Suc_1
tff(fact_297_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) ).

% mult_Suc_right
tff(fact_298_pred__less__length__list,axiom,
    ! [Dega: nat,Xb: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
         => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb),aa(nat,option(nat),some(nat),Mia),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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_299_pred__lesseq__max,axiom,
    ! [Dega: nat,Xb: nat,Maa: nat,Mia: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa)
       => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb),aa(nat,option(nat),some(nat),Mia),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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_300_add__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(nat,option(nat),some(nat),Xb)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% add_shift
tff(fact_301_mul__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xb),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),Xb)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% mul_shift
tff(fact_302_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_303_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_304_nat_Oinject,axiom,
    ! [X23: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X23) = aa(nat,nat,suc,Y2) )
    <=> ( X23 = Y2 ) ) ).

% nat.inject
tff(fact_305_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_306_real__divide__square__eq,axiom,
    ! [R: 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),R),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),R),R)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),R) ).

% real_divide_square_eq
tff(fact_307_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(Ta),Xb,Sx) ) ) ).

% succ_correct
tff(fact_308_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(Ta),Xb,Sx) ) ) ).

% pred_correct
tff(fact_309_lessI,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Nb)) ).

% lessI
tff(fact_310_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb)) ) ).

% Suc_mono
tff(fact_311_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_less_eq
tff(fact_312_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,suc,Mb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb) ) ).

% Suc_le_mono
tff(fact_313_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ).

% add_Suc_right
tff(fact_314_nat__add__left__cancel__less,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_315_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% diff_Suc_Suc
tff(fact_316_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),K) ).

% Suc_diff_diff
tff(fact_317_nat__add__left__cancel__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_318_diff__diff__cancel,axiom,
    ! [I2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_319_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_320_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) )
    <=> ( ( Mb = one_one(nat) )
        & ( Nb = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_321_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) = one_one(nat) )
    <=> ( ( Mb = one_one(nat) )
        & ( Nb = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_322_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X2: A] :
              ( ! [Y5: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y5)),aa(A,B,F2,X2))
                 => aa(A,$o,P,Y5) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct
tff(fact_323_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X2: A] :
              ( ! [Y5: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y5)),aa(A,B,F2,X2))
                 => aa(A,$o,P,Y5) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct_rule
tff(fact_324_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xb) = aa(nat,nat,suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_325_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa(nat,nat,suc,Nb) ).

% n_not_Suc_n
tff(fact_326_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_327_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_not_refl
tff(fact_328_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_329_less__not__refl3,axiom,
    ! [S3: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S3),Ta)
     => ( S3 != Ta ) ) ).

% less_not_refl3
tff(fact_330_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_irrefl_nat
tff(fact_331_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% nat_less_induct
tff(fact_332_infinite__descent,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P,N)
         => ? [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
              & ~ aa(nat,$o,P,M2) ) )
     => aa(nat,$o,P,Nb) ) ).

% infinite_descent
tff(fact_333_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_334_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V3: fun(A,nat),Xb: A] :
      ( ! [X2: A] :
          ( ~ aa(A,$o,P,X2)
         => ? [Y5: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V3,Y5)),aa(A,nat,V3,X2))
              & ~ aa(A,$o,P,Y5) ) )
     => aa(A,$o,P,Xb) ) ).

% infinite_descent_measure
tff(fact_335_le__refl,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Nb) ).

% le_refl
tff(fact_336_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),K) ) ) ).

% le_trans
tff(fact_337_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% eq_imp_le
tff(fact_338_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_339_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb) ) ).

% nat_le_linear
tff(fact_340_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y4: nat] :
            ( aa(nat,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),B2) )
       => ? [X2: nat] :
            ( aa(nat,$o,P,X2)
            & ! [Y5: nat] :
                ( aa(nat,$o,P,Y5)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y5),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_341_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,nat,size_size(A),Xb) != aa(A,nat,size_size(A),Y) )
         => ( Xb != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_342_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I2),K)),J) ).

% diff_commute
tff(fact_343_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_344_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
     => ( ( K != aa(nat,nat,suc,I2) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_345_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_lessD
tff(fact_346_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_347_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( ( aa(nat,nat,suc,Mb) != Nb )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_348_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_349_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb)) ) ).

% less_SucI
tff(fact_350_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Nb)
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
            & aa(nat,$o,P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_351_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_352_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb)) ) ).

% not_less_eq
tff(fact_353_All__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Nb)
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
           => aa(nat,$o,P,I4) ) ) ) ).

% All_less_Suc
tff(fact_354_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Mb)
    <=> ? [M3: nat] :
          ( ( Mb = aa(nat,nat,suc,M3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),M3) ) ) ).

% Suc_less_eq2
tff(fact_355_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_356_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_357_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),K) ) ) ).

% less_trans_Suc
tff(fact_358_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( ! [I3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I3),aa(nat,nat,suc,I3))
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I3),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I3),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J) ) ) ) ).

% less_Suc_induct
tff(fact_359_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( ! [I3: nat] :
            ( ( J = aa(nat,nat,suc,I3) )
           => aa(nat,$o,P,I3) )
       => ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I3))
               => aa(nat,$o,P,I3) ) )
         => aa(nat,$o,P,I2) ) ) ) ).

% strict_inc_induct
tff(fact_360_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_361_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% Suc_leD
tff(fact_362_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ( Mb = aa(nat,nat,suc,Nb) ) ) ) ).

% le_SucE
tff(fact_363_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb)) ) ).

% le_SucI
tff(fact_364_Suc__le__D,axiom,
    ! [Nb: nat,M4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),M4)
     => ? [M: nat] : M4 = aa(nat,nat,suc,M) ) ).

% Suc_le_D
tff(fact_365_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        | ( Mb = aa(nat,nat,suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_366_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_367_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_368_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M2)),N)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N) )
     => aa(nat,$o,P,Nb) ) ).

% full_nat_induct
tff(fact_369_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,$o,P,Mb)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_370_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R2: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,X2),X2)
       => ( ! [X2: nat,Y4: nat,Z4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R2,X2),Y4)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R2,Y4),Z4)
               => aa(nat,$o,aa(nat,fun(nat,$o),R2,X2),Z4) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R2,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R2,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_371_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,suc,Nb)) ).

% add_Suc_shift
tff(fact_372_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ).

% add_Suc
tff(fact_373_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_374_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_375_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_376_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_377_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Mb = Nb ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_378_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( ( Mb != Nb )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_379_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I2: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,I3)),aa(nat,nat,F2,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F2,I2)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_380_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_381_trans__less__add2,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),J)) ) ).

% trans_less_add2
tff(fact_382_trans__less__add1,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Mb)) ) ).

% trans_less_add1
tff(fact_383_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_less_mono1
tff(fact_384_not__add__less2,axiom,
    ! [J: nat,I2: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),I2) ).

% not_add_less2
tff(fact_385_not__add__less1,axiom,
    ! [I2: nat,J: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),I2) ).

% not_add_less1
tff(fact_386_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_387_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K) ) ).

% add_lessD1
tff(fact_388_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K: nat,I2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I2)) ) ) ).

% zero_induct_lemma
tff(fact_389_Suc__mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_390_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
    <=> ? [K3: nat] : Nb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K3) ) ).

% nat_le_iff_add
tff(fact_391_trans__le__add2,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),J)) ) ).

% trans_le_add2
tff(fact_392_trans__le__add1,axiom,
    ! [I2: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Mb)) ) ).

% trans_le_add1
tff(fact_393_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_394_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_395_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
     => ? [N: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) ) ).

% le_Suc_ex
tff(fact_396_add__leD2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% add_leD2
tff(fact_397_add__leD1,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% add_leD1
tff(fact_398_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ).

% le_add2
tff(fact_399_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)) ).

% le_add1
tff(fact_400_add__leE,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),Nb)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% add_leE
tff(fact_401_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_402_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_403_eq__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_404_le__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_405_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_406_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),L)) ) ).

% diff_le_mono
tff(fact_407_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_408_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),A2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_409_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Mb)) ) ).

% diff_le_mono2
tff(fact_410_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_411_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_412_diff__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% diff_cancel2
tff(fact_413_Nat_Odiff__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% Nat.diff_cancel
tff(fact_414_le__cube,axiom,
    ! [Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Mb))) ).

% le_cube
tff(fact_415_le__square,axiom,
    ! [Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Mb)) ).

% le_square
tff(fact_416_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_417_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_418_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_419_add__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ).

% add_mult_distrib2
tff(fact_420_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: 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),Mb),Nb)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K)) ).

% add_mult_distrib
tff(fact_421_nat__mult__1,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),Nb) = Nb ).

% nat_mult_1
tff(fact_422_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),one_one(nat)) = Nb ).

% nat_mult_1_right
tff(fact_423_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K)) ).

% diff_mult_distrib
tff(fact_424_diff__mult__distrib2,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ).

% diff_mult_distrib2
tff(fact_425_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,Xb: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xb) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_426_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,N3)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_427_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,Mb: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Mb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_428_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,N3)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_429_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,N))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N3)),aa(nat,A,F2,Nb)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_430_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_431_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_le_eq
tff(fact_432_dec__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,P,I2)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),J)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) )
         => aa(nat,$o,P,J) ) ) ) ).

% dec_induct
tff(fact_433_inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),J)
               => ( aa(nat,$o,P,aa(nat,nat,suc,N))
                 => aa(nat,$o,P,N) ) ) )
         => aa(nat,$o,P,I2) ) ) ) ).

% inc_induct
tff(fact_434_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_435_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_436_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_437_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_438_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_439_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ? [K2: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2)) ) ).

% less_imp_Suc_add
tff(fact_440_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
    <=> ? [K3: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K3)) ) ).

% less_iff_Suc_add
tff(fact_441_less__add__Suc2,axiom,
    ! [I2: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I2))) ).

% less_add_Suc2
tff(fact_442_less__add__Suc1,axiom,
    ! [I2: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),Mb))) ).

% less_add_Suc1
tff(fact_443_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ~ ! [Q3: nat] : Nb != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q3)) ) ).

% less_natE
tff(fact_444_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_445_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),aa(nat,nat,suc,Mb)) ).

% diff_less_Suc
tff(fact_446_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),Nb) ).

% Suc_eq_plus1_left
tff(fact_447_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_448_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_449_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_450_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),Mb: nat,K: nat] :
      ( ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,M)),aa(nat,nat,F2,N)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,Mb)),K)),aa(nat,nat,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K))) ) ).

% mono_nat_linear_lb
tff(fact_451_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_452_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_453_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) ).

% mult_Suc
tff(fact_454_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_455_less__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_456_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),A2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C2)) ) ) ).

% diff_less_mono
tff(fact_457_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_458_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J) ) ).

% less_diff_conv
tff(fact_459_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I2)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)) ) ).

% le_diff_conv
tff(fact_460_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_461_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_462_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_463_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_464_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_465_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I2)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)) ) ) ).

% less_diff_conv2
tff(fact_466_member__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r(Ta,Xb)),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,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% member_bound_height
tff(fact_467_succ__less__length__list,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list(vEBT_VEBT),Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Xb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
         => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = 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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_468_succ__greatereq__min,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Xb)
       => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = 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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_469_vebt__pred_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
        aa(nat,option(nat),some(nat),Maa),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,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),aa(num,num,bit0,one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,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),aa(num,num,bit0,one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                  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),aa(num,num,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),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                  $let(
                    pr: option(nat),
                    pr:= vEBT_vebt_pred(Summarya,h),
                    $ite(
                      pr = none(nat),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb),aa(nat,option(nat),some(nat),Mia),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),aa(num,num,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),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_470_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa(real,real,aa(real,fun(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),aa(num,num,bit0,one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% real_average_minus_first
tff(fact_471_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa(real,real,aa(real,fun(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),aa(num,num,bit0,one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% real_average_minus_second
tff(fact_472_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_473_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_474_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
        aa(nat,option(nat),some(nat),Mia),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,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),aa(num,num,bit0,one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,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),aa(num,num,bit0,one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  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),aa(num,num,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),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Summarya,h),
                    $ite(sc = 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),aa(num,num,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),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_475_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeLista,Summarya),Xb) = 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),Xb),Xb)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_476_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aa(vEBT_VEBT,fun(nat,fun(nat,$o)),Ta),Xb)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_477_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ab(vEBT_VEBT,fun(nat,fun(nat,$o)),Ta),Xb)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_478_less__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
    <=> vEBT_VEBT_less(aa(nat,option(nat),some(nat),Xb),aa(nat,option(nat),some(nat),Y)) ) ).

% less_shift
tff(fact_479_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = none(nat) )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_480_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = none(nat) )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_481_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_482_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_483_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_ac(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_484_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_485_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_486_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_487_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_488_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_489_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_490_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,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(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_491_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,aa(A,fun(A,A),minus_minus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(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_492_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,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(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_493_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,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(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_494_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ad(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_495_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Mb: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),Nb)) ) ) ) ).

% less_1_mult
tff(fact_496_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))) ) ).

% less_add_one
tff(fact_497_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_498_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K: A,Nb: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_499_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_500_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_501_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: 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)),D2) )
        <=> ( 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),minus_minus(A),A2),B2)),E)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_502_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: 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)),D2) )
        <=> ( 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),minus_minus(A),B2),A2)),E)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_503_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) ) ).

% square_diff_square_factored
tff(fact_504_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,Xb,Y)
    <=> ( aa(set(nat),$o,member(nat,Y),Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
        & ! [X: nat] :
            ( aa(set(nat),$o,member(nat,X),Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),X)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_505_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,Xb,Y)
    <=> ( aa(set(nat),$o,member(nat,Y),Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xb)
        & ! [X: nat] :
            ( aa(set(nat),$o,member(nat,X),Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Xb)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_506_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),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,aa(A,fun(A,A),minus_minus(A),B2),A2)),E)),D2)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_507_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),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,aa(A,fun(A,A),minus_minus(A),A2),B2)),E)),C2)),D2) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_508_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),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,aa(A,fun(A,A),minus_minus(A),B2),A2)),E)),D2)) ) ) ).

% less_add_iff2
tff(fact_509_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),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,aa(A,fun(A,A),minus_minus(A),A2),B2)),E)),C2)),D2) ) ) ).

% less_add_iff1
tff(fact_510_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),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),Xb),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_511_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_512_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_513_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xb,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),aa(num,num,bit0,one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xb,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),aa(num,num,bit0,one2)))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
                  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,TreeLista),h)))),
                    $let(
                      maxlow: option(nat),
                      maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                        $ite(
                          ( ( maxlow != none(nat) )
                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                          $let(
                            sc: option(nat),
                            sc:= vEBT_vebt_succ(Summarya,h),
                            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(Summarya,h))),one_one(nat))),
                              $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_514_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,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),aa(num,num,bit0,one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,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),aa(num,num,bit0,one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  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,TreeLista),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summarya,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_515_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_516_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),Mb)),aa(num,extended_enat,numeral_numeral(extended_enat),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Mb)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% enat_ord_number(1)
tff(fact_517_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),Mb)),aa(num,extended_enat,numeral_numeral(extended_enat),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),Mb)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% enat_ord_number(2)
tff(fact_518_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),aa(num,num,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_519_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) ).

% Suc_double_not_eq_double
tff(fact_520_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb) != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_521_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_522_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_523_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_524_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_525_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),aa(num,num,bit0,V))),aa(num,int,numeral_numeral(int),aa(num,num,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_526_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Z),Y)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xb),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Y),Z)) = aa(extended_enat,extended_enat,aa(extended_enat,fun(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),Xb),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_527_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X4) ) ).

% linordered_field_no_lb
tff(fact_528_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X4: A] :
        ? [X_13: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),X_13) ) ).

% linordered_field_no_ub
tff(fact_529_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_530_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_531_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_532_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: 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),Xb),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),Xb),W)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_533_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: 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),Xb),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),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W)) ) ).

% times_divide_times_eq
tff(fact_534_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_535_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,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(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_536_succ_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c2(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% succ'_bound_height
tff(fact_537_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),B2) ) ) ).

% discrete
tff(fact_538_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_539_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_540_succ__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c(Ta,Xb)),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,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2))))))) ) ).

% succ_bound_height
tff(fact_541_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q2: A,R: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R),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),aa(num,num,bit0,one2))),Q2)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(num,A,numeral_numeral(A),L))),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),aa(num,num,bit0,one2))),Q2)),R)) ) ).

% divmod_step_eq
tff(fact_542_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_543_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_right
tff(fact_544_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_545_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_left
tff(fact_546_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,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_547_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_548_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: 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),Mia),Maa)),aa(nat,nat,suc,V),TreeLista,Vc),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_549_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_550_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_551_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_min_max
tff(fact_552_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_553_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_554_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_555_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_556_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_left
tff(fact_557_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_right
tff(fact_558_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,A)),Uv2: option(A)] : Xb != 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)),V2: A] : Xb != 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),V2)),none(A)))
       => ~ ! [F3: fun(A,fun(A,A)),A4: A,B3: A] : Xb != 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))),F3),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_559_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,$o)),Uv2: option(A)] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,$o)),V2: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),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),V2)),none(A)))
       => ~ ! [F3: fun(A,fun(A,$o)),X2: A,Y4: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),F3),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),X2)),aa(A,option(A),some(A),Y4))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_560_xor__num_Ocases,axiom,
    ! [Xb: product_prod(num,num)] :
      ( ( Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N))
       => ( ! [N: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N))
         => ( ! [M: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M)),one2)
           => ( ! [M: num,N: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M)),aa(num,num,bit0,N))
             => ( ! [M: num,N: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M)),aa(num,num,bit1,N))
               => ( ! [M: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M)),one2)
                 => ( ! [M: num,N: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M)),aa(num,num,bit0,N))
                   => ~ ! [M: num,N: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_561_mult_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)) ) ).

% mult.left_commute
tff(fact_562_mult_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) ) ).

% mult.commute
tff(fact_563_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_564_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_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)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_565_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xb: A] :
          ( ( one_one(A) = Xb )
        <=> ( Xb = one_one(A) ) ) ) ).

% one_reorient
tff(fact_566_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_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)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_567_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( ( I2 = J )
            & ( K = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_568_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_569_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_570_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_571_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_572_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_573_add_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) ) ).

% add.commute
tff(fact_574_add_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)) ) ).

% add.left_commute
tff(fact_575_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_576_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_577_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) ) ).

% diff_right_commute
tff(fact_578_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( ( A2 = B2 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_579_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_580_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( ( I2 = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_581_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_582_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ).

% add_mono
tff(fact_583_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_584_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_585_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_586_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_587_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_588_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_589_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_590_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( ( I2 = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_591_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_592_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ).

% add_strict_mono
tff(fact_593_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_594_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),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_595_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_left
tff(fact_596_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_right
tff(fact_597_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_598_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_right_mono
tff(fact_599_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_left_mono
tff(fact_600_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_mono
tff(fact_601_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_602_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_603_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_604_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_605_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_606_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_607_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,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_608_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(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_609_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(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_610_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,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_611_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_612_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,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_613_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_614_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,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_615_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_616_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_617_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_618_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_619_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_620_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( ( aa(A,A,aa(A,fun(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_621_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_622_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(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_623_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_624_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C2) = aa(A,A,aa(A,fun(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_625_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_626_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(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_627_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),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_628_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2)) ) ) ).

% le_add_diff
tff(fact_629_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),A2) = B2 ) ) ) ).

% diff_add
tff(fact_630_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% le_diff_eq
tff(fact_631_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_le_eq
tff(fact_632_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_less_eq
tff(fact_633_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% less_diff_eq
tff(fact_634_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list(vEBT_VEBT),Vd: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeLista,Vd),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_635_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tree,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
       => ( vEBT_V5719532721284313246member(Tree,Xb)
          | vEBT_VEBT_membermima(Tree,Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_636_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
    <=> ( vEBT_V5719532721284313246member(Ta,Xb)
        | vEBT_VEBT_membermima(Ta,Xb) ) ) ).

% both_member_options_def
tff(fact_637_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = 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),Va2,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : Xb = 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,V2),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_638_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeLista: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeLista,S3),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_639_insert__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,Xb)),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,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2))))))) ) ).

% insert_bound_height
tff(fact_640_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,B3: $o] : Xb = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),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),h)))),
                                        $let(
                                          maxlow: option(nat),
                                          maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                            $ite(
                                              ( ( maxlow != none(nat) )
                                              & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                              $let(
                                                sc: option(nat),
                                                sc:= vEBT_vebt_succ(Summary,h),
                                                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,h))),one_one(nat))),
                                                  $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                      one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_641_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_leaf
tff(fact_642_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,Uw2: $o] : Xb = vEBT_Leaf((A4),(Uw2))
           => ( ( Xaa = 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: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A4),(B3)) )
               => ( ? [Va3: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va3))
                 => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $ite((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),Va2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = 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,Va3)),TreeList,Summary) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,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(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),one_one(nat))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                          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,bit0,one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                            $ite(
                                              ( ( minlow != none(nat) )
                                              & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                              $let(
                                                pr: option(nat),
                                                pr:= vEBT_vebt_pred(Summary,h),
                                                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,h))),one_one(nat))),
                                                  $ite(pr = 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),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                        one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_643_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_644_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xb: A] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I3)) )
     => ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_645_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_0_not
tff(fact_646_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_647_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% deg_not_0
tff(fact_648_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_649_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
    <=> ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ).

% deg1Leaf
tff(fact_650_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
     => ? [A4: $o,B3: $o] : Ta = vEBT_Leaf((A4),(B3)) ) ).

% deg_1_Leaf
tff(fact_651_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( Nb = one_one(nat) )
       => ? [A4: $o,B3: $o] : Ta = vEBT_Leaf((A4),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_652_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => vEBT_invar_vebt(vEBT_vebt_buildup(Nb),Nb) ) ).

% buildup_gives_valid
tff(fact_653_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X222)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_654_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),zero_zero(A))
        <=> ( Nb = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_655_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb)
        <=> ( Nb = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_656_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_657_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_658_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_659_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_660_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_661_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_662_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_663_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_664_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_665_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_666_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_667_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_668_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_669_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_670_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% diff_self
tff(fact_671_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_0_right
tff(fact_672_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_diff
tff(fact_673_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_zero
tff(fact_674_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_684_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% neq0_conv
tff(fact_685_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_686_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),A2) ).

% bot_nat_0.extremum
tff(fact_687_le0,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% le0
tff(fact_688_Nat_Oadd__0__right,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),zero_zero(nat)) = Mb ).

% Nat.add_0_right
tff(fact_689_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) = zero_zero(nat) )
    <=> ( ( Mb = zero_zero(nat) )
        & ( Nb = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_690_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),Nb) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_691_diff__self__eq__0,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Mb) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_692_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) = zero_zero(nat) )
    <=> ( ( Mb = zero_zero(nat) )
        | ( Nb = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_693_mult__0__right,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_694_mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_695_mult__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_696_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_697_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_698_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_699_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_700_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_701_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_702_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_703_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_704_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel2
tff(fact_705_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel1
tff(fact_706_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_707_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_708_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_709_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% diff_gt_0_iff_gt
tff(fact_710_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_711_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_712_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_713_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_714_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: 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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_715_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_716_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_717_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_718_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_719_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_720_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_721_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: 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)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_722_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_723_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_724_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: 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)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% div_mult_mult1_if
tff(fact_725_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_726_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_727_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_728_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_729_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_730_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_731_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_732_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_733_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_734_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_735_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_736_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,Nb)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_737_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_738_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_739_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_740_zero__less__Suc,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,Nb)) ).

% zero_less_Suc
tff(fact_741_add__gr__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% add_gr_0
tff(fact_742_less__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),one_one(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_one
tff(fact_743_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) )
    <=> ( ( Mb = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_744_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Mb = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_745_div__by__Suc__0,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,suc,zero_zero(nat))) = Mb ).

% div_by_Suc_0
tff(fact_746_zero__less__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% zero_less_diff
tff(fact_747_mult__less__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_748_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_749_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_750_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = zero_zero(nat) ) ) ).

% div_less
tff(fact_751_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_752_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_753_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Mb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xb),Mb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Mb = zero_zero(nat) )
        | ( Xb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_754_power__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_755_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_756_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = $ite(K = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_757_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_758_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_759_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_760_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_761_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_762_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_763_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_764_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_765_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)) )
        <=> $ite(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,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_766_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 )
        <=> $ite(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)),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_767_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_768_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_769_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_770_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_771_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_772_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_773_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% power_eq_0_iff
tff(fact_774_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))) = Nb ) ) ).

% Suc_pred
tff(fact_775_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) ) ) ).

% one_le_mult_iff
tff(fact_776_mult__le__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_777_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_778_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_779_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Mb)),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_780_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_781_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_782_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_783_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_784_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_785_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),aa(num,num,bit0,one2))) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_786_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_787_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) = Nb ) ) ).

% Suc_diff_1
tff(fact_788_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),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_789_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),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_790_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
            <=> ( Xb = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_791_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_792_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb) ) ) ) ) ).

% power_decreasing_iff
tff(fact_793_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_794_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: 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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_795_option_Osize__neq,axiom,
    ! [A: $tType,Xb: option(A)] : aa(option(A),nat,size_size(option(A)),Xb) != zero_zero(nat) ).

% option.size_neq
tff(fact_796_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf((X21),(X222))) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_797_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A2),(B2)),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A2),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_798_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_799_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_800_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A2),(B2)),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(Xb = 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_801_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B3: $o,X2: nat] : Xb != 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))),X2)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : Xb != 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)),V2: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT,X2: nat] : Xb != 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,V2),TreeList,S2)),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_802_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_803_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Nb) = $ite(Nb = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% power_0_left
tff(fact_804_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Nb) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_805_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A2),(B2))),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A2),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_806_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: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_807_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_808_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] : Xb != 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))),D3)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg2: nat] : Xb != 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,Deg,TreeList,Summary)),Deg2) ) ).

% VEBT_internal.valid'.cases
tff(fact_809_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_810_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((Uu),(Uv)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_811_vebt__insert_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A2),(B2)),Xb) = $ite(
        Xb = zero_zero(nat),
        vEBT_Leaf($true,(B2)),
        $ite(Xb = one_one(nat),vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_812_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv)),zero_zero(nat)) = none(nat) ).

% vebt_pred.simps(1)
tff(fact_813_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uu),(B2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_814_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_815_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) ).

% zero_le
tff(fact_816_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),zero_zero(A)) ) ).

% less_numeral_extra(3)
tff(fact_817_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D1)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D22)
           => ? [E2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E2),D1)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E2),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_818_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ( Nb != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb) ) ) ).

% gr_zeroI
tff(fact_819_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Nb),zero_zero(A)) ) ).

% not_less_zero
tff(fact_820_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Mb: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb)
         => ( Nb != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_821_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Nb)
        <=> ( Nb != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_822_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),Nb) ) ).

% zero_neq_numeral
tff(fact_823_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_824_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_825_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_826_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_827_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_828_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_829_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_830_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_831_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_832_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_833_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Nb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_834_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_835_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero(nat) != aa(nat,nat,suc,X23) ).

% nat.distinct(1)
tff(fact_836_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_837_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_838_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa(nat,nat,suc,X23) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_839_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_840_nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P,N)
           => aa(nat,$o,P,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P,Nb) ) ) ).

% nat_induct
tff(fact_841_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Mb: nat,Nb: nat] :
      ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X2),zero_zero(nat))
     => ( ! [Y4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y4))
       => ( ! [X2: nat,Y4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X2),Y4)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X2)),aa(nat,nat,suc,Y4)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_842_zero__induct,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N))
           => aa(nat,$o,P,N) )
       => aa(nat,$o,P,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_843_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa(nat,nat,suc,Mb) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_844_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_neq_Suc
tff(fact_845_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_not_Suc
tff(fact_846_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ? [M: nat] : Nb = aa(nat,nat,suc,M) ) ).

% not0_implies_Suc
tff(fact_847_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ( ( Xb != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va3: nat] : Xb != aa(nat,nat,suc,aa(nat,nat,suc,Va3)) ) ) ).

% vebt_buildup.cases
tff(fact_848_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,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero(nat),Ts,S3),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_849_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_850_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% gr0I
tff(fact_851_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_852_not__less0,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% not_less0
tff(fact_853_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_zeroE
tff(fact_854_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( Nb != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_855_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( ~ aa(nat,$o,P,N)
             => ? [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                  & ~ aa(nat,$o,P,M2) ) ) )
       => aa(nat,$o,P,Nb) ) ) ).

% infinite_descent0
tff(fact_856_infinite__descent0__measure,axiom,
    ! [A: $tType,V3: fun(A,nat),P: fun(A,$o),Xb: A] :
      ( ! [X2: A] :
          ( ( aa(A,nat,V3,X2) = zero_zero(nat) )
         => aa(A,$o,P,X2) )
     => ( ! [X2: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V3,X2))
           => ( ~ aa(A,$o,P,X2)
             => ? [Y5: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V3,Y5)),aa(A,nat,V3,X2))
                  & ~ aa(A,$o,P,Y5) ) ) )
       => aa(A,$o,P,Xb) ) ) ).

% infinite_descent0_measure
tff(fact_857_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% less_eq_nat.simps(1)
tff(fact_858_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_859_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_860_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),zero_zero(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_861_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) = Mb )
     => ( Nb = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_862_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),Nb) = Nb ).

% plus_nat.add_0
tff(fact_863_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv)),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_864_minus__nat_Odiff__0,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),zero_zero(nat)) = Mb ).

% minus_nat.diff_0
tff(fact_865_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb) = zero_zero(nat) )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_866_mult__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),Nb) = zero_zero(nat) ).

% mult_0
tff(fact_867_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
    <=> ( ( K = zero_zero(nat) )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_868_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_869_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_870_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ae(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_871_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : vEBT_T_s_u_c_c(vEBT_Leaf((Uu),(B2)),zero_zero(nat)) = 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_872_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_T_m_i_n_t(vEBT_Leaf((A2),(B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((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_873_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,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_874_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ) ).

% power_strict_mono
tff(fact_875_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((A2),(Uw)),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_876_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(Uw)),aa(nat,nat,suc,zero_zero(nat))) = $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ).

% vebt_pred.simps(2)
tff(fact_877_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A2),(B2))) = $ite(
        (A2),
        aa(nat,option(nat),some(nat),zero_zero(nat)),
        $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ).

% vebt_mint.simps(1)
tff(fact_878_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu),(B2)),zero_zero(nat)) = $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ).

% vebt_succ.simps(1)
tff(fact_879_vebt__maxt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A2),(B2))) = $ite(
        (B2),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_maxt.simps(1)
tff(fact_880_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_le_numeral
tff(fact_881_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_882_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_883_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_884_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),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_885_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),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_886_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_887_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_888_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),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_889_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_890_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),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_891_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_892_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) )
         => aa(A,$o,aa(A,fun(A,$o),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_893_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),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_894_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_895_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),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_896_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_897_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_898_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),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_899_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_900_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_less_numeral
tff(fact_901_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_902_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_903_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_904_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing
tff(fact_905_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing
tff(fact_906_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing2
tff(fact_907_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing2
tff(fact_908_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),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_909_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_910_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_911_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_912_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_neg_neg
tff(fact_913_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),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_914_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_less_0_iff
tff(fact_915_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_916_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_917_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_pos_pos
tff(fact_918_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_919_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_920_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos
tff(fact_921_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos2
tff(fact_922_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_923_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_924_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_925_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),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_926_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_927_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_928_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),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_929_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_930_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),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_931_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% less_numeral_extra(1)
tff(fact_932_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one
tff(fact_933_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),zero_zero(A)) ) ).

% not_one_less_zero
tff(fact_934_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_935_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% pos_add_strict
tff(fact_936_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_937_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_pos
tff(fact_938_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_939_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_940_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_le_0_iff
tff(fact_941_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),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_942_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_943_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_944_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_945_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_946_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_947_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_948_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_949_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_950_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),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_951_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_952_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_953_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_less_0_iff
tff(fact_954_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_pos_pos
tff(fact_955_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_956_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_957_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_neg_neg
tff(fact_958_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono
tff(fact_959_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% zero_le_power
tff(fact_960_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% zero_less_power
tff(fact_961_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_962_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_963_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_964_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_965_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) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_966_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 )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_967_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),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),Xb),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_968_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_969_length__pos__if__in__set,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_970_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_971_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
            & aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_972_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
    <=> ? [M5: nat] : Nb = aa(nat,nat,suc,M5) ) ).

% gr0_conv_Suc
tff(fact_973_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_974_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ? [M: nat] : Nb = aa(nat,nat,suc,M) ) ).

% gr0_implies_Suc
tff(fact_975_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,suc,Nb))
    <=> ( ( Mb = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( Mb = aa(nat,nat,suc,J3) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_976_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) )
    <=> ( ( ( Mb = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Nb = zero_zero(nat) ) )
        | ( ( Mb = zero_zero(nat) )
          & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_977_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( Mb = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Nb = zero_zero(nat) ) )
        | ( ( Mb = zero_zero(nat) )
          & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_978_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_979_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_980_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
            & ! [I: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K2)
               => ~ aa(nat,$o,P,I) )
            & aa(nat,$o,P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_981_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ? [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_982_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_983_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_984_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ) ).

% mult_less_mono1
tff(fact_985_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ) ).

% mult_less_mono2
tff(fact_986_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_987_nat__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_988_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_989_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)) = zero_zero(nat) ).

% diff_add_0
tff(fact_990_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) )
     => ( ( Nb = one_one(nat) )
        | ( Mb = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_991_nat__power__less__imp__less,axiom,
    ! [I2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),I2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),Nb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_992_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,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B3: $o,X2: nat] : Xb != 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))),X2)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X2: nat] : Xb != 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)),X2)
       => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,X2: nat] : Xb != 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)),V2),zero_zero(nat),Uy2,Uz2)),X2)
         => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X2: nat] : Xb != 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X2)
           => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),X2) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_993_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,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B3: $o,X2: nat] : Xb != 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))),X2)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X2: nat] : Xb != 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)),X2)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X2: nat] : Xb != 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)),X2)
         => ( ! [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != 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,V2)),TreeList,Summary)),X2)
           => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),X2) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_994_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,B3: $o] : Xb != 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: $o,Uw2: $o,N: nat] : Xb != 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,N))
       => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va2: nat] : Xb != 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)),Va2)
         => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve: nat] : Xb != 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)),V2),zero_zero(nat),Vc2,Vd2)),Ve)
           => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : Xb != 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Vi)
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),X2) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_995_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o] : Xb != 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: $o,Uw2: $o] : Xb != 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: $o,B3: $o,Va3: nat] : Xb != 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,Va3)))
         => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va2: vEBT_VEBT,Vb2: nat] : Xb != 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,Va2)),Vb2)
           => ( ! [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : Xb != 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)),V2),zero_zero(nat),Vd2,Ve)),Vf)
             => ( ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : Xb != 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Vj)
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),X2) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.cases
tff(fact_996_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xb != 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] : Xb != 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)
       => ( ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT,X2: nat] : Xb != 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),Mi),Ma)),zero_zero(nat),Va2,Vb2)),X2)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,X2: nat] : Xb != 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),Mi),Ma)),aa(nat,nat,suc,V2),TreeList,Vc2)),X2)
           => ~ ! [V2: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT,X2: nat] : Xb != 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,V2),TreeList,Vd2)),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_997_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts,S3),Xb) = vEBT_Node(Info,zero_zero(nat),Ts,S3) ).

% vebt_insert.simps(2)
tff(fact_998_vebt__pred_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        (B2),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_pred.simps(3)
tff(fact_999_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,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_1000_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A2),(B2)),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
        $ite(Xb = 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_1001_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv),(Uw)),aa(nat,nat,suc,Nb)) = none(nat) ).

% vebt_succ.simps(2)
tff(fact_1002_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uv),(Uw)),aa(nat,nat,suc,Nb)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_1003_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_T_s_u_c_c(vEBT_Leaf((Uv),(Uw)),aa(nat,nat,suc,Nb)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_1004_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1005_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1006_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1007_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1008_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1009_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1010_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1011_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1012_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1013_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1014_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1015_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1016_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1017_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1018_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% field_le_epsilon
tff(fact_1019_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_1020_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_pos
tff(fact_1021_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_1022_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_nonneg
tff(fact_1023_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_1024_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_1025_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb)),Xb) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1026_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)),Xb) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1027_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),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_1028_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),A2) ) ) ) ).

% mult_left_le
tff(fact_1029_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_le
tff(fact_1030_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less
tff(fact_1031_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less2
tff(fact_1032_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_1033_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1034_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1035_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1036_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1037_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),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_1038_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_positive
tff(fact_1039_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1040_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1041_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( Xb != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1042_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),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),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_1043_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),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_1044_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% power_less_imp_less_base
tff(fact_1045_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1046_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% divide_less_eq
tff(fact_1047_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_1048_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_1049_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1050_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1051_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_1052_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1053_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1054_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),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_1055_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),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_1056_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1057_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1058_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_1059_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) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B2,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1060_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) )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2),aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1061_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: 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),Xb),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1062_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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),Xb),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1063_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: 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),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_1064_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xb: 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),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_1065_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: 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),Xb),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),Xb),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_1066_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: 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)) = $ite(Z = zero_zero(A),A2,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_1067_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: 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) = $ite(Z = zero_zero(A),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_1068_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1069_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% power_le_imp_le_base
tff(fact_1070_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_1071_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_1072_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_T_m_a_x_t(vEBT_Leaf((A2),(B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((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_1073_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1074_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),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),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),Y)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1075_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),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),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),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_1076_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(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_1077_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_1078_num_Osize_I5_J,axiom,
    ! [X23: num] : aa(num,nat,size_size(num),aa(num,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_1079_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B4)
    <=> ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
         => aa(set(A),$o,member(A,X),B4) ) ) ).

% subset_code(1)
tff(fact_1080_Ex__list__of__length,axiom,
    ! [A: $tType,Nb: nat] :
    ? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1081_neq__if__length__neq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) != aa(list(A),nat,size_size(list(A)),Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1082_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Nb)
            & ! [I: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K2)
               => ~ aa(nat,$o,P,I) )
            & aa(nat,$o,P,aa(nat,nat,suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_1083_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,P,N)
               => aa(nat,$o,P,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_1084_num_Osize_I6_J,axiom,
    ! [X32: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_1085_diff__Suc__less,axiom,
    ! [Nb: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,I2))),Nb) ) ).

% diff_Suc_less
tff(fact_1086_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_1087_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_1088_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_1089_nat__diff__split,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D4: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
           => aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split
tff(fact_1090_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D4: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
              & ~ aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_1091_nat__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_1092_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Mb)) ) ) ).

% div_le_mono2
tff(fact_1093_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_1094_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),K)) ) ).

% power_gt_expt
tff(fact_1095_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = Mb )
      <=> ( Nb = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1096_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_1097_nat__one__le__power,axiom,
    ! [I2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),Nb)) ) ).

% nat_one_le_power
tff(fact_1098_nat__mult__div__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_1099_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Q2)),Nb)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_1100_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),Xb) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) ).

% vebt_insert.simps(3)
tff(fact_1101_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,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)),Xb) ).

% vebt_member.simps(3)
tff(fact_1102_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xb) = Y )
     => ( ! [A4: $o] :
            ( ? [B3: $o] : Xb = vEBT_Leaf((A4),(B3))
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((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] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = 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)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_1103_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_1104_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] : Xb != vEBT_Leaf((A4),(B3))
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb != 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)),Ux2,Uy2,Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_1105_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] :
      vEBT_T_p_r_e_d(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((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_1106_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [Z4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z4)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z4),Xb)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1107_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1108_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1109_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1110_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1111_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1112_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1113_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1114_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1115_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_1116_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1117_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),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_1118_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_1119_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1120_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1121_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_1122_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1123_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1124_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),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_1125_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1126_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1127_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xb: A,A2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),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),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1128_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( Y != $ite(
                  (A4),
                  aa(nat,option(nat),some(nat),zero_zero(nat)),
                  $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = 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)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_1129_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1130_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1131_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( Y != $ite(
                  (B3),
                  aa(nat,option(nat),some(nat),one_one(nat)),
                  $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = 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)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_1132_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),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_1133_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),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),Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% power_Suc_less
tff(fact_1134_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),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_1135_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))),A2) ) ) ) ).

% power_Suc_le_self
tff(fact_1136_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_1137_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1138_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N2: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_decreasing
tff(fact_1139_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),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_1140_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% self_le_power
tff(fact_1141_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% one_less_power
tff(fact_1142_numeral__2__eq__2,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% numeral_2_eq_2
tff(fact_1143_pos2,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% pos2
tff(fact_1144_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,Nb: nat,Mb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_diff
tff(fact_1145_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_1146_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_1147_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1148_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb) = $ite(Mb = zero_zero(nat),Nb,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))),Nb))) ).

% add_eq_if
tff(fact_1149_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_1150_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_1151_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Q2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1152_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_1153_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_1154_split__div,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I4)),J3) )
               => aa(nat,$o,P,I4) ) ) ) ) ) ).

% split_div
tff(fact_1155_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb) = $ite(Mb = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))),Nb))) ).

% mult_eq_if
tff(fact_1156_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList,S2)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_1157_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList,S2)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_1158_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList,S2)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_1159_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,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,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_1160_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),Ve2: vEBT_VEBT,Vf2: 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,Ve2),Vf2) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_1161_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Va: list(vEBT_VEBT),Vb: vEBT_VEBT,Xb: 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),Mia),Maa)),zero_zero(nat),Va,Vb),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1162_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc,Vd),Ve2) = none(nat) ).

% vebt_succ.simps(4)
tff(fact_1163_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: 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,Ve2),Vf2) = none(nat) ).

% vebt_pred.simps(5)
tff(fact_1164_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve2: 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),Vc,Vd),Ve2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_1165_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve2: 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),Vc,Vd),Ve2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_1166_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xb: A,A2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),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),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1167_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1168_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1169_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_1170_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ).

% half_gt_zero
tff(fact_1171_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R: A,S3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R),S3)
             => aa(A,$o,aa(A,fun(A,$o),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),R),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U))),S3))),V) ) ) ) ) ).

% scaling_mono
tff(fact_1172_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_1173_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => ( Xb = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1174_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),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),aa(num,num,bit0,one2)))) ) ).

% zero_le_power2
tff(fact_1175_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_1176_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1177_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1178_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,Mb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1179_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),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,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1180_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
    <=> ( ( Nb = zero_zero(nat) )
        | ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_1181_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))
     => ( ( Nb = zero_zero(nat) )
        | ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_1182_nat__induct2,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,P,N)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct2
tff(fact_1183_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P2: A,Mb: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),Mb) = $ite(Mb = zero_zero(nat),one_one(A),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,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1184_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( 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,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))),A2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) ) ) ) ).

% power_minus_mult
tff(fact_1185_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_1186_split__div_H,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)),Mb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q4)))
            & aa(nat,$o,P,Q4) ) ) ) ).

% split_div'
tff(fact_1187_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),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: 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)),Vh2,Vi2),Vj2) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_1188_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2),Vi2) = none(nat) ).

% vebt_succ.simps(5)
tff(fact_1189_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: 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)),Vh2,Vi2),Vj2) = none(nat) ).

% vebt_pred.simps(6)
tff(fact_1190_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),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: 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)),Vg2,Vh2),Vi2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_1191_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),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: 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)),Vg2,Vh2),Vi2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_1192_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_1193_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A))
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_1194_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% sum_power2_ge_zero
tff(fact_1195_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
        <=> ( ( Xb != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_1196_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A)) ) ).

% not_sum_power2_lt_zero
tff(fact_1197_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),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),aa(num,num,bit0,one2))),Nb))) ) ).

% zero_le_even_power'
tff(fact_1198_nat__bit__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P,N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ) )
       => ( ! [N: nat] :
              ( aa(nat,$o,P,N)
             => aa(nat,$o,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_1199_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1200_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% div_2_gt_zero
tff(fact_1201_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(list(A),$o,P,Ys2) )
         => aa(list(A),$o,P,Xs2) )
     => aa(list(A),$o,P,Xs) ) ).

% length_induct
tff(fact_1202_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,Xb: 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),Xb) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_1203_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xb) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((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] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = 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)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_1204_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1205_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),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),aa(num,num,bit0,one2))),Nb)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_1206_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1207_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(Xb,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1208_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Summary: vEBT_VEBT] : Xb = 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,Va3)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                        $false,
                        $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_1209_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = 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),Va2,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = 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,V2),TreeList,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_1210_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = 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),Va2,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = 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,V2),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd2)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_1211_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),Vc: vEBT_VEBT,Xb: 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,Vc),Xb) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_1212_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => (Y) )
         => ( ( ? [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)
             => (Y) )
           => ( ( ? [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = 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,Va3)),TreeList,Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_1213_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)
           => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = 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,Va3)),TreeList,Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_1214_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,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: 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)),TreeLista,Summarya),Xb) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_1215_insersimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_13)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insersimp
tff(fact_1216_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => aa(A,$o,aa(A,fun(A,$o),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),Xb),Y)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_1217_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xb,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                $ite(Xaa = 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] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) )
         => ( ( ? [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)
             => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = 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,Va3)),TreeList,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                          $ite(
                            Xaa = Mi,
                            one_one(nat),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Ma,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                        one_one(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,bit0,aa(num,num,bit0,one2))))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),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),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),one_one(nat)) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_1218_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A5: $o,B5: $o] : A1 = vEBT_Leaf((A5),(B5))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N4) )
            & vEBT_invar_vebt(Summary2,N4)
            & ( 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),aa(num,num,bit0,one2))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
            & ! [X: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N4) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N4))
            & ( 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),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(nat,nat,suc,N4)) )
            & ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
            & ! [X: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N4) )
            & vEBT_invar_vebt(Summary2,N4)
            & ( 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),aa(num,num,bit0,one2))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4))
               => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_1)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4))
                 => ( ( ( vEBT_VEBT_high(Ma2,N4) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N4)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N4) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X,N4)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT,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)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N4) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N4))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(nat,nat,suc,N4)) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)))
               => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_1)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)))
                 => ( ( ( vEBT_VEBT_high(Ma2,N4) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N4)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N4) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X,N4)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1219_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: $o,B3: $o] : A1 = vEBT_Leaf((A4),(B3))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X4: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => vEBT_invar_vebt(X4,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),aa(num,num,bit0,one2))),M) )
                     => ( ( M = N )
                       => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                         => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_12)
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X4: vEBT_VEBT] :
                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => vEBT_invar_vebt(X4,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),aa(num,num,bit0,one2))),M) )
                       => ( ( M = aa(nat,nat,suc,N) )
                         => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                           => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_12)
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X4: vEBT_VEBT] :
                          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_invar_vebt(X4,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),aa(num,num,bit0,one2))),M) )
                         => ( ( M = N )
                           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                             => ( ! [I: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))
                                   => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),X_1)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X4: vEBT_VEBT] :
                                        ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) )
                                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I: nat] :
                                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))
                                             => ( ( ( vEBT_VEBT_high(Ma,N) = I )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),vEBT_VEBT_low(Ma,N)) )
                                                & ! [X4: nat] :
                                                    ( ( ( vEBT_VEBT_high(X4,N) = I )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),vEBT_VEBT_low(X4,N)) )
                                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X4)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: 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),Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X4: vEBT_VEBT] :
                            ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_invar_vebt(X4,N) )
                       => ( vEBT_invar_vebt(Summary,M)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
                           => ( ( M = aa(nat,nat,suc,N) )
                             => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                               => ( ! [I: nat] :
                                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))
                                     => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),X_1)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X4: vEBT_VEBT] :
                                          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                         => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_12) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))
                                               => ( ( ( vEBT_VEBT_high(Ma,N) = I )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),vEBT_VEBT_low(Ma,N)) )
                                                  & ! [X4: nat] :
                                                      ( ( ( vEBT_VEBT_high(X4,N) = I )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),vEBT_VEBT_low(X4,N)) )
                                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X4)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_1220_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1221_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
         => ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P,I4),X_1) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I4),aa(nat,A,nth(A,Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_1222_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1223_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,B3: $o] : Xb = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            one_one(nat),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      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),h),l)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
                                  one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_1224_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,Vd2)
               => ( Y != none(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            aa(nat,option(nat),some(nat),Mi),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                      $let(
                                        sc: option(nat),
                                        sc:= vEBT_vebt_succ(Summary,h),
                                        $ite(sc = 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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                  none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_1225_vebt__pred_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw2: $o] : Xb = vEBT_Leaf((A4),(Uw2))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A4),(B3)) )
               => ( ? [Va3: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va3))
                 => ( Y != $ite(
                        (B3),
                        aa(nat,option(nat),some(nat),one_one(nat)),
                        $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va2)
               => ( Y != none(nat) ) )
             => ( ( ? [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve)
                 => ( Y != none(nat) ) )
               => ( ( ? [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = 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,Va3)),TreeList,Summary) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                              aa(nat,option(nat),some(nat),Ma),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                        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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                        $let(
                                          pr: option(nat),
                                          pr:= vEBT_vebt_pred(Summary,h),
                                          $ite(
                                            pr = none(nat),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xaa),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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                    none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_1226_nth__mem,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => aa(set(A),$o,member(A,aa(nat,A,nth(A,Xs),Nb)),aa(list(A),set(A),set2(A),Xs)) ) ).

% nth_mem
tff(fact_1227_list__ball__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P,X2) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_1228_in__set__conv__nth,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_1229_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xb,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = 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,Va3)),TreeList,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                          $ite(
                            Xaa = Mi,
                            zero_zero(nat),
                            $ite(
                              Xaa = Ma,
                              zero_zero(nat),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                zero_zero(nat),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                  zero_zero(nat),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xaa)
                                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma) ),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),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),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(nat)) ),
                                    zero_zero(nat) ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_1230_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,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          Xb = Mia,
          zero_zero(nat),
          $ite(
            Xb = Maa,
            zero_zero(nat),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),
              zero_zero(nat),
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),Xb),
                zero_zero(nat),
                $ite(
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb)
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Maa) ),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(Xb,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),aa(num,num,bit0,one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(Xb,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),aa(num,num,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_1231_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(Ta)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),one_one(nat)))) ) ).

% inrange
tff(fact_1232_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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% set_bit_0
tff(fact_1233_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = 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: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( 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,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = 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)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,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)),V2),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = 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,Va3)),TreeList,Summary) )
                       => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),
                                        $ite(
                                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),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),h)))),
                                            $let(
                                              maxlow: option(nat),
                                              maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                                $ite(
                                                  ( ( maxlow != none(nat) )
                                                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                                  $let(
                                                    sc: option(nat),
                                                    sc:= vEBT_vebt_succ(Summary,h),
                                                    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,h))),one_one(nat))),
                                                      $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                          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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_1234_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = 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: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = 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_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: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A4),(B3)) )
                 => ! [Va3: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va3)) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite((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_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,Va3)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,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_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va2)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve) )
                     => ( ( 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)),V2),zero_zero(nat),Vd2,Ve)),Xaa)) ) )
                 => ( ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xb = 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,Va3)),TreeList,Summary) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                    one_one(nat),
                                    $let(
                                      l: nat,
                                      l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                      $let(
                                        h: nat,
                                        h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,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(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),one_one(nat))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                            $let(
                                              minlow: option(nat),
                                              minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                              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,bit0,one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                                $ite(
                                                  ( ( minlow != none(nat) )
                                                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                                  $let(
                                                    pr: option(nat),
                                                    pr:= vEBT_vebt_pred(Summary,h),
                                                    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,h))),one_one(nat))),
                                                      $ite(pr = 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),aa(num,num,bit0,aa(num,num,bit0,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                            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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_1235_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),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: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( 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,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = 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)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,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)),V2),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = 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,Va3)),TreeList,Summary) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                aa(nat,option(nat),some(nat),Mi),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                          $let(
                                            sc: option(nat),
                                            sc:= vEBT_vebt_succ(Summary,h),
                                            $ite(sc = 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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                      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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_1236_vebt__pred_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = 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: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),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: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A4),(B3)) )
                 => ! [Va3: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va3)) )
                     => ( ( Y = $ite(
                              (B3),
                              aa(nat,option(nat),some(nat),one_one(nat)),
                              $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),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,Va3)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va2) )
                   => ( ( 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,Va2)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vd2,Ve) )
                     => ( ( 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)),V2),zero_zero(nat),Vd2,Ve)),Xaa)) ) )
                 => ( ! [V2: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xb = 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,Va3)),TreeList,Summary) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                  aa(nat,option(nat),some(nat),Ma),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                            $let(
                                              pr: option(nat),
                                              pr:= vEBT_vebt_pred(Summary,h),
                                              $ite(
                                                pr = none(nat),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xaa),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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                        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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_1237_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_1238_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Ta)
     => ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_13) ) ).

% not_min_Null_member
tff(fact_1239_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_minNull(Ta)
     => ~ aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ).

% min_Null_member
tff(fact_1240_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none(nat) )
     => vEBT_VEBT_minNull(Ta) ) ).

% minminNull
tff(fact_1241_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Ta)
     => ( vEBT_vebt_mint(Ta) = none(nat) ) ) ).

% minNullmin
tff(fact_1242_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),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_1243_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),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_1244_idiff__0__right,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Nb),zero_zero(extended_enat)) = Nb ).

% idiff_0_right
tff(fact_1245_idiff__0,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),zero_zero(extended_enat)),Nb) = zero_zero(extended_enat) ).

% idiff_0
tff(fact_1246_not__real__square__gt__zero,axiom,
    ! [Xb: real] :
      ( ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Xb))
    <=> ( Xb = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1247_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_1248_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_1249_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_1250_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: 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)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1251_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: 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)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1252_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),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,aa(int,fun(int,int),minus_minus(int),K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1253_split__zdiv,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_1254_enat__0__less__mult__iff,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),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),Mb),Nb))
    <=> ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Mb)
        & aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_1255_iadd__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Mb),Nb) = zero_zero(extended_enat) )
    <=> ( ( Mb = zero_zero(extended_enat) )
        & ( Nb = zero_zero(extended_enat) ) ) ) ).

% iadd_is_0
tff(fact_1256_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1257_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1258_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull(vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1259_all__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),Nb)
         => aa(nat,$o,P,M5) )
    <=> ! [X: nat] :
          ( aa(set(nat),$o,member(nat,X),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less
tff(fact_1260_ex__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),Nb)
          & aa(nat,$o,P,M5) )
    <=> ? [X: nat] :
          ( aa(set(nat),$o,member(nat,X),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less
tff(fact_1261_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : ~ vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_1262_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1263_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_1264_realpow__pos__nth2,axiom,
    ! [A2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ? [R3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),aa(nat,nat,suc,Nb)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1265_real__arch__pow__inv,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1266_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((A2),(B2)),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_1267_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1268_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1269_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X2)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X2),Nb) = A2 )
            & ! [Y5: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y5)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y5),Nb) = A2 ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1270_realpow__pos__nth,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R3),Nb) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1271_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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_1272_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_1273_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1274_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,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_1275_int__power__div__base,axiom,
    ! [Mb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(int,$o,aa(int,fun(int,$o),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),Mb)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1276_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,Xb: 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),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_1277_insertsimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(Ta)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insertsimp
tff(fact_1278_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),Vc: vEBT_VEBT,Xb: 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,Vc),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_1279_member__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r2(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% member_bound_height'
tff(fact_1280_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = 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: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( 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,N))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = 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)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Vc2,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)),V2),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V2: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = 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,Va3)),TreeList,Summary) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          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),h),l)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_1281_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                      $ite(Xaa = 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))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,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)),Xaa)) ) )
           => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,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)),V2),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                              $ite(
                                Xaa = Mi,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    Xaa = Ma,
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                            one_one(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,bit0,aa(num,num,bit0,one2))))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),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),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_1282_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = 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))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = 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)),Xaa)) ) )
           => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),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)),V2),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Mi,
                                zero_zero(nat),
                                $ite(
                                  Xaa = Ma,
                                  zero_zero(nat),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                    zero_zero(nat),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                      zero_zero(nat),
                                      $ite(
                                        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xaa)
                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma) ),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),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),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_1283_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xb,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(Xaa = 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] : Xb = 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] : Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary)
               => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = 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,Va3)),TreeList,Summary) )
                   => ( 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,aa(num,num,bit0,aa(num,num,bit0,one2)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                            $let(
                              h: nat,
                              h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $ite(
                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                                & ~ ( ( Xaa = Mi )
                                    | ( Xaa = 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),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
                                one_one(nat) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_1284_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,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = 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,aa(num,num,bit0,aa(num,num,bit0,one2)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Mia,Xb),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(xn,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),aa(num,num,bit0,one2)))),
            $ite(
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
              & ~ ( ( Xb = Mia )
                  | ( Xb = Maa ) ) ),
              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,TreeLista),h),vEBT_VEBT_low(xn,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),aa(num,num,bit0,one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)))),
                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_T_i_n_s_e_r_t(Summarya,h),one_one(nat))),
              one_one(nat) ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_1285_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( 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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = 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))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = 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)),Xaa)) )
           => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),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)),V2),zero_zero(nat),Uy2,Uz2)),Xaa)) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ~ 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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_1286_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ 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))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ (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)),Xaa)) ) )
           => ( ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2) )
                 => ( ~ (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)),V2),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ (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)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) )
                       => ~ 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_1287_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( 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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = 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))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = 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)),Xaa)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList,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,V2),TreeList,S2)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_1288_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1289_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_1290_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ).

% zle_add1_eq_le
tff(fact_1291_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ).

% zle_diff1_eq
tff(fact_1292_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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)),R))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1293_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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)),R))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1294_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R: 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)),R) = 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) )
     => ( aa(int,$o,aa(int,fun(int,$o),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))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1295_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R: 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)),R) = 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) )
     => ( aa(int,$o,aa(int,fun(int,$o),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))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B6)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B6),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1296_q__pos__lemma,axiom,
    ! [B6: int,Q5: int,R4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B6)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B6)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_1297_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_1298_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => aa(int,$o,aa(int,fun(int,$o),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_1299_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_1300_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_1301_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_1302_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_1303_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_1304_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_1305_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1306_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Mb)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb) = one_one(int) )
      <=> ( ( Mb = one_one(int) )
          & ( Nb = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1307_imult__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),Mb),Nb) = zero_zero(extended_enat) )
    <=> ( ( Mb = zero_zero(extended_enat) )
        | ( Nb = zero_zero(extended_enat) ) ) ) ).

% imult_is_0
tff(fact_1308_set__bit__greater__eq,axiom,
    ! [K: int,Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K)) ).

% set_bit_greater_eq
tff(fact_1309_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,aa(int,fun(int,int),minus_minus(int),Z1),Z22)) = aa(int,int,aa(int,fun(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_1310_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,aa(int,fun(int,int),minus_minus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(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_1311_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_1312_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_1313_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ).

% add1_zle_eq
tff(fact_1314_int__gr__induct,axiom,
    ! [K: int,I2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I2)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I3)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I2) ) ) ) ).

% int_gr_induct
tff(fact_1315_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1316_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
     => aa(int,$o,aa(int,fun(int,$o),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_1317_int__ge__induct,axiom,
    ! [K: int,I2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
     => ( aa(int,$o,P,K)
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I3)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I2) ) ) ) ).

% int_ge_induct
tff(fact_1318_int__less__induct,axiom,
    ! [I2: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I2) ) ) ) ).

% int_less_induct
tff(fact_1319_int__le__induct,axiom,
    ! [I2: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
     => ( aa(int,$o,P,K)
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I2) ) ) ) ).

% int_le_induct
tff(fact_1320_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I2: int] :
      ( aa(int,$o,P,K)
     => ( ! [I3: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I3)
           => ( aa(int,$o,P,I3)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I2) ) ) ) ).

% int_induct
tff(fact_1321_minNull__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_N_u_l_l(Ta)),one_one(nat)) ).

% minNull_bound
tff(fact_1322_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: $o] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf((Uu),$true)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_1323_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv: $o] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf($true,(Uv))) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_1324_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf($false,$false)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_1325_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),Vc: 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,Vc)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_1326_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_1327_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,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xb) = Y )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => ( Y != one_one(nat) ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => ( Y != one_one(nat) ) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ( Y != one_one(nat) ) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)
                 => ( Y != one_one(nat) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_1328_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( 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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = 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))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = 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,Va3)),TreeList,Summary) )
               => ( 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_1329_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ 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))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ (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)),Xaa)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList,S2) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) )
                   => ~ 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,V2),TreeList,S2)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1330_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( 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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = 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))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList,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,V2),TreeList,S2)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_1331_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite(Xaa = 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))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = 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)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = 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)),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,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,V2)),TreeList,Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( ( 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,aa(num,num,bit0,aa(num,num,bit0,one2)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                                    & ~ ( ( Xaa = Mi )
                                        | ( Xaa = 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),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                                      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary,h),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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_1332_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (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),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (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))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ (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)),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = 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),Va2,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ 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),Mi),Ma)),zero_zero(nat),Va2,Vb2)),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = 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,V2),TreeList,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) )
                     => ~ 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),Mi),Ma)),aa(nat,nat,suc,V2),TreeList,Vc2)),Xaa)) ) )
               => ~ ! [V2: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) )
                       => ~ 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,V2),TreeList,Vd2)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_1333_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( 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),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = 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))),Xaa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = 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)),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = 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),Va2,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),Mi),Ma)),zero_zero(nat),Va2,Vb2)),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = 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,V2),TreeList,Vc2) )
                   => ( 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),Mi),Ma)),aa(nat,nat,suc,V2),TreeList,Vc2)),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,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,V2),TreeList,Vd2)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_1334_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xb,Xaa) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xb = 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] : Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = 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,Va3)),TreeList,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = 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),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
                              one_one(nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_1335_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( 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),Xb),Xaa))
       => ( ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( Xb = 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),Va2,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),Mi),Ma)),zero_zero(nat),Va2,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( Xb = 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,V2),TreeList,Vc2) )
               => ( 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),Mi),Ma)),aa(nat,nat,suc,V2),TreeList,Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,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,V2),TreeList,Vd2)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_1336_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_1337_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1338_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_1339_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I2),set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_1340_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L2: A,H2: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L2,H2) )
        <=> ( ( ( L = L2 )
              & ( H = H2 ) )
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
              & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L2),H2) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_1341_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff
tff(fact_1342_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((A2),(B2)),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_1343_bounded__Max__nat,axiom,
    ! [P: fun(nat,$o),Xb: nat,M6: nat] :
      ( aa(nat,$o,P,Xb)
     => ( ! [X2: nat] :
            ( aa(nat,$o,P,X2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),M6) )
       => ~ ! [M: nat] :
              ( aa(nat,$o,P,M)
             => ~ ! [X4: nat] :
                    ( aa(nat,$o,P,X4)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),M) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1344_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F3: fun(nat,fun(A,A)),A4: nat,B3: nat,Acc: A] : Xb != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1345_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,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero(nat),Ts,S3),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_1346_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,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_1347_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,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: 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)),TreeLista,Summarya),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_1348_insersimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_13)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,Y)),one_one(nat)) ) ) ).

% insersimp'
tff(fact_1349_insertsimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(Ta)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,L)),one_one(nat)) ) ) ).

% insertsimp'
tff(fact_1350_insert_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% insert'_bound_height
tff(fact_1351_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_1352_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,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,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),aa(num,num,bit0,one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            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,TreeLista),h),vEBT_VEBT_low(xn,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),aa(num,num,bit0,one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_T_i_n_s_e_r_t2(Summarya,h),one_one(nat))),
            one_one(nat) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_1353_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,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = 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))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = 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)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = 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)),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary) )
                   => ( ( 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,V2)),TreeList,Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = 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),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_1354_cppi,axiom,
    ! [D5: int,P: fun(int,$o),P3: fun(int,$o),A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),X2)
           => ( aa(int,$o,P,X2)
            <=> aa(int,$o,P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb2: int] :
                      ( aa(set(int),$o,member(int,Xb2),A3)
                     => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa(int,$o,P3,X2)
              <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D5))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X: int] :
                    ( aa(set(int),$o,member(int,X),set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P3,X) )
                | ? [X: int] :
                    ( aa(set(int),$o,member(int,X),set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa2: int] :
                        ( aa(set(int),$o,member(int,Xa2),A3)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa2),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_1355_cpmi,axiom,
    ! [D5: int,P: fun(int,$o),P3: fun(int,$o),B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( ? [Z2: int] :
          ! [X2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X2),Z2)
           => ( aa(int,$o,P,X2)
            <=> aa(int,$o,P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb2: int] :
                      ( aa(set(int),$o,member(int,Xb2),B4)
                     => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D5)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa(int,$o,P3,X2)
              <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D5))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X: int] :
                    ( aa(set(int),$o,member(int,X),set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P3,X) )
                | ? [X: int] :
                    ( aa(set(int),$o,member(int,X),set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa2: int] :
                        ( aa(set(int),$o,member(int,Xa2),B4)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_1356_bset_I6_J,axiom,
    ! [D5: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5)),Ta) ) ) ) ).

% bset(6)
tff(fact_1357_bset_I8_J,axiom,
    ! [D5: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int))),B4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5)) ) ) ) ) ).

% bset(8)
tff(fact_1358_aset_I6_J,axiom,
    ! [D5: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A3)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X4),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_1359_aset_I8_J,axiom,
    ! [D5: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A3)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5)) ) ) ) ).

% aset(8)
tff(fact_1360_bset_I3_J,axiom,
    ! [D5: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int))),B4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X4 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_1361_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_1362_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X4) ) ) ).

% minf(7)
tff(fact_1363_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Ta) ) ) ).

% minf(5)
tff(fact_1364_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
         => ( X4 != Ta ) ) ) ).

% minf(4)
tff(fact_1365_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
         => ( X4 != Ta ) ) ) ).

% minf(3)
tff(fact_1366_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z2: A] :
            ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
             => ( aa(A,$o,P,X2)
              <=> aa(A,$o,P3,X2) ) )
         => ( ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
               => ( aa(A,$o,Q,X2)
                <=> aa(A,$o,Q6,X2) ) )
           => ? [Z4: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
               => ( ( aa(A,$o,P,X4)
                    | aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P3,X4)
                    | aa(A,$o,Q6,X4) ) ) ) ) ) ) ).

% minf(2)
tff(fact_1367_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z2: A] :
            ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
             => ( aa(A,$o,P,X2)
              <=> aa(A,$o,P3,X2) ) )
         => ( ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
               => ( aa(A,$o,Q,X2)
                <=> aa(A,$o,Q6,X2) ) )
           => ? [Z4: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
               => ( ( aa(A,$o,P,X4)
                    & aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P3,X4)
                    & aa(A,$o,Q6,X4) ) ) ) ) ) ) ).

% minf(1)
tff(fact_1368_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_1369_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X4) ) ) ).

% pinf(7)
tff(fact_1370_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Ta) ) ) ).

% pinf(5)
tff(fact_1371_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
         => ( X4 != Ta ) ) ) ).

% pinf(4)
tff(fact_1372_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
         => ( X4 != Ta ) ) ) ).

% pinf(3)
tff(fact_1373_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z2: A] :
            ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
             => ( aa(A,$o,P,X2)
              <=> aa(A,$o,P3,X2) ) )
         => ( ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
               => ( aa(A,$o,Q,X2)
                <=> aa(A,$o,Q6,X2) ) )
           => ? [Z4: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
               => ( ( aa(A,$o,P,X4)
                    | aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P3,X4)
                    | aa(A,$o,Q6,X4) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_1374_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z2: A] :
            ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
             => ( aa(A,$o,P,X2)
              <=> aa(A,$o,P3,X2) ) )
         => ( ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
               => ( aa(A,$o,Q,X2)
                <=> aa(A,$o,Q6,X2) ) )
           => ? [Z4: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
               => ( ( aa(A,$o,P,X4)
                    & aa(A,$o,Q,X4) )
                <=> ( aa(A,$o,P3,X4)
                    & aa(A,$o,Q6,X4) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_1375_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X4) ) ) ).

% minf(8)
tff(fact_1376_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Ta) ) ) ).

% minf(6)
tff(fact_1377_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X4) ) ) ).

% pinf(8)
tff(fact_1378_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Ta) ) ) ).

% pinf(6)
tff(fact_1379_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D5: A,Q: fun(A,$o)] :
          ( ! [X2: A,K2: A] :
              ( aa(A,$o,P,X2)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D5))) )
         => ( ! [X2: A,K2: A] :
                ( aa(A,$o,Q,X2)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D5))) )
           => ! [X4: A,K4: A] :
                ( ( aa(A,$o,P,X4)
                  | aa(A,$o,Q,X4) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5)))
                  | aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1380_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D5: A,Q: fun(A,$o)] :
          ( ! [X2: A,K2: A] :
              ( aa(A,$o,P,X2)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D5))) )
         => ( ! [X2: A,K2: A] :
                ( aa(A,$o,Q,X2)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D5))) )
           => ! [X4: A,K4: A] :
                ( ( aa(A,$o,P,X4)
                  & aa(A,$o,Q,X4) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5)))
                  & aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1381_aset_I2_J,axiom,
    ! [D5: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A3)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A3)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X4)
                | aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5)) ) ) ) ) ) ).

% aset(2)
tff(fact_1382_aset_I1_J,axiom,
    ! [D5: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),A3)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),A3)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X4)
                & aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5)) ) ) ) ) ) ).

% aset(1)
tff(fact_1383_bset_I2_J,axiom,
    ! [D5: int,B4: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B4)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B4)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D5)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X4)
                | aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5)) ) ) ) ) ) ).

% bset(2)
tff(fact_1384_bset_I1_J,axiom,
    ! [D5: int,B4: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( aa(set(int),$o,member(int,Xb2),B4)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( aa(set(int),$o,member(int,Xa),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( aa(set(int),$o,member(int,Xb2),B4)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D5)) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X4)
                & aa(int,$o,Q,X4) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5)) ) ) ) ) ) ).

% bset(1)
tff(fact_1385_plusinfinity,axiom,
    ! [D2: int,P3: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int,K2: int] :
            ( aa(int,$o,P3,X2)
          <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z2: int] :
            ! [X2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z2),X2)
             => ( aa(int,$o,P,X2)
              <=> aa(int,$o,P3,X2) ) )
         => ( ? [X_12: int] : aa(int,$o,P3,X_12)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_1386_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int,K2: int] :
            ( aa(int,$o,P1,X2)
          <=> aa(int,$o,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z2: int] :
            ! [X2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X2),Z2)
             => ( aa(int,$o,P,X2)
              <=> aa(int,$o,P1,X2) ) )
         => ( ? [X_12: int] : aa(int,$o,P1,X_12)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_1387_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int] :
            ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X4: int] :
              ( aa(int,$o,P,X4)
             => aa(int,$o,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),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1388_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int] :
            ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X4: int] :
              ( aa(int,$o,P,X4)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1389_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X2: int,K2: int] :
            ( aa(int,$o,P,X2)
          <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [X_1: int] : aa(int,$o,P,X_1)
        <=> ? [X: int] :
              ( aa(set(int),$o,member(int,X),set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,P,X) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1390_aset_I7_J,axiom,
    ! [D5: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A3)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5)) ) ) ) ).

% aset(7)
tff(fact_1391_aset_I5_J,axiom,
    ! [D5: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( aa(set(int),$o,member(int,Ta),A3)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_1392_aset_I4_J,axiom,
    ! [D5: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( aa(set(int),$o,member(int,Ta),A3)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X4 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_1393_aset_I3_J,axiom,
    ! [D5: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( aa(set(int),$o,member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int))),A3)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),A3)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X4 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_1394_bset_I7_J,axiom,
    ! [D5: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( aa(set(int),$o,member(int,Ta),B4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X4)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5)) ) ) ) ) ).

% bset(7)
tff(fact_1395_bset_I5_J,axiom,
    ! [D5: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5)),Ta) ) ) ) ).

% bset(5)
tff(fact_1396_bset_I4_J,axiom,
    ! [D5: int,Ta: int,B4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( aa(set(int),$o,member(int,Ta),B4)
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( aa(set(int),$o,member(int,Xb3),B4)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X4 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_1397_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4) ) ).

% Diff_eq_empty_iff
tff(fact_1398_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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% unset_bit_0
tff(fact_1399_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [A4: real,B3: real,C3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A4),B3)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B3),C3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),B3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),C3)
                 => aa(real,$o,aa(real,fun(real,$o),P,A4),C3) ) ) ) )
       => ( ! [X2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
               => ? [D6: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                    & ! [A4: real,B3: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),X2)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B3)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B3),A4)),D6) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A4),B3) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_1400_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,Nb: nat] :
      ( ! [N: nat] : aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,H,N)
     => ( ( aa(nat,A,F2,zero_zero(nat)) = G )
       => ( aa(nat,A,F2,Nb) = $ite(Nb = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_1401_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] :
          unique8689654367752047608divmod(A,aa(num,num,bit0,Mb),aa(num,num,bit1,Nb)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb),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,bit0,Mb))),unique1321980374590559556d_step(A,aa(num,num,bit1,Nb),unique8689654367752047608divmod(A,aa(num,num,bit0,Mb),aa(num,num,bit0,aa(num,num,bit1,Nb))))) ) ).

% divmod_algorithm_code(7)
tff(fact_1402_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] :
          unique8689654367752047608divmod(A,aa(num,num,bit1,Mb),aa(num,num,bit1,Nb)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb),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,Mb))),unique1321980374590559556d_step(A,aa(num,num,bit1,Nb),unique8689654367752047608divmod(A,aa(num,num,bit1,Mb),aa(num,num,bit0,aa(num,num,bit1,Nb))))) ) ).

% divmod_algorithm_code(8)
tff(fact_1403_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1404_DiffI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,member(A,C2),A3)
     => ( ~ aa(set(A),$o,member(A,C2),B4)
       => aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) ) ) ).

% DiffI
tff(fact_1405_Diff__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))
    <=> ( aa(set(A),$o,member(A,C2),A3)
        & ~ aa(set(A),$o,member(A,C2),B4) ) ) ).

% Diff_iff
tff(fact_1406_Diff__idemp,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) ).

% Diff_idemp
tff(fact_1407_Diff__cancel,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),A3) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1408_empty__Diff,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A3) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1409_Diff__empty,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),bot_bot(set(A))) = A3 ).

% Diff_empty
tff(fact_1410_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_1411_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_1412_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num] : unique8689654367752047608divmod(A,Mb,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),Mb)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_1413_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit0,Nb)) = 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_1414_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,Nb)) = 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_1415_minus__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),minus_minus(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B4))) ).

% minus_set_def
tff(fact_1416_set__diff__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_af(set(A),fun(set(A),fun(A,$o)),A3),B4)) ).

% set_diff_eq
tff(fact_1417_DiffE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))
     => ~ ( aa(set(A),$o,member(A,C2),A3)
         => aa(set(A),$o,member(A,C2),B4) ) ) ).

% DiffE
tff(fact_1418_DiffD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))
     => aa(set(A),$o,member(A,C2),A3) ) ).

% DiffD1
tff(fact_1419_DiffD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,member(A,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))
     => ~ aa(set(A),$o,member(A,C2),B4) ) ).

% DiffD2
tff(fact_1420_unset__bit__less__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K)),K) ).

% unset_bit_less_eq
tff(fact_1421_double__diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C5)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1422_Diff__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),A3) ).

% Diff_subset
tff(fact_1423_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),D5: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D5),B4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),D5)) ) ) ).

% Diff_mono
tff(fact_1424_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4)
     => ? [B3: A] : aa(set(A),$o,member(A,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)) ) ).

% psubset_imp_ex_mem
tff(fact_1425_empty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ag(A,$o)) ).

% empty_def
tff(fact_1426_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] :
          unique8689654367752047608divmod(A,Mb,Nb) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb),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),Mb)),unique1321980374590559556d_step(A,Nb,unique8689654367752047608divmod(A,Mb,aa(num,num,bit0,Nb)))) ) ).

% divmod_divmod_step
tff(fact_1427_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% mult_less_iff1
tff(fact_1428_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1429_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R))
        <=> ( R = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_1430_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),R))
       => 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),aa(num,num,bit0,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_1431_low__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_low(Xb,Nb) = modulo_modulo(nat,Xb,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% low_def
tff(fact_1432_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A3: set(nat),B4: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Z)
     => ( vEBT_VEBT_max_in_set(A3,Z)
       => ( finite_finite2(nat,B4)
         => ( ( A3 = B4 )
           => ? [X_13: nat] : vEBT_is_succ_in_set(A3,Xb,X_13) ) ) ) ) ).

% obtain_set_succ
tff(fact_1433_obtain__set__pred,axiom,
    ! [Z: nat,Xb: nat,A3: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z),Xb)
     => ( vEBT_VEBT_min_in_set(A3,Z)
       => ( finite_finite2(nat,A3)
         => ? [X_13: nat] : vEBT_is_pred_in_set(A3,Xb,X_13) ) ) ) ).

% obtain_set_pred
tff(fact_1434_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A4: A,B3: B,C3: C,D3: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D3))) ).

% prod_cases4
tff(fact_1435_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3))))] :
      ~ ! [A4: A,B3: B,C3: C,D3: D,E2: E3] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E3))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E3))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E3)))),A4),aa(product_prod(C,product_prod(D,E3)),product_prod(B,product_prod(C,product_prod(D,E3))),aa(B,fun(product_prod(C,product_prod(D,E3)),product_prod(B,product_prod(C,product_prod(D,E3)))),product_Pair(B,product_prod(C,product_prod(D,E3))),B3),aa(product_prod(D,E3),product_prod(C,product_prod(D,E3)),aa(C,fun(product_prod(D,E3),product_prod(C,product_prod(D,E3))),product_Pair(C,product_prod(D,E3)),C3),aa(E3,product_prod(D,E3),aa(D,fun(E3,product_prod(D,E3)),product_Pair(D,E3),D3),E2)))) ).

% prod_cases5
tff(fact_1436_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))))] :
      ~ ! [A4: A,B3: B,C3: C,D3: D,E2: E3,F3: F] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),A4),aa(product_prod(C,product_prod(D,product_prod(E3,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E3,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E3,F)))),B3),aa(product_prod(D,product_prod(E3,F)),product_prod(C,product_prod(D,product_prod(E3,F))),aa(C,fun(product_prod(D,product_prod(E3,F)),product_prod(C,product_prod(D,product_prod(E3,F)))),product_Pair(C,product_prod(D,product_prod(E3,F))),C3),aa(product_prod(E3,F),product_prod(D,product_prod(E3,F)),aa(D,fun(product_prod(E3,F),product_prod(D,product_prod(E3,F))),product_Pair(D,product_prod(E3,F)),D3),aa(F,product_prod(E3,F),aa(E3,fun(F,product_prod(E3,F)),product_Pair(E3,F),E2),F3))))) ).

% prod_cases6
tff(fact_1437_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => finite_finite2(nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_1438_pred__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_13: nat] : vEBT_is_pred_in_set(Xs,A2,X_13)
     => ( finite_finite2(nat,Xs)
       => ~ ? [X4: nat] :
              ( aa(set(nat),$o,member(nat,X4),Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),A2) ) ) ) ).

% pred_none_empty
tff(fact_1439_succ__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_13: nat] : vEBT_is_succ_in_set(Xs,A2,X_13)
     => ( finite_finite2(nat,Xs)
       => ~ ? [X4: nat] :
              ( aa(set(nat),$o,member(nat,X4),Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),X4) ) ) ) ).

% succ_none_empty
tff(fact_1440_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6) )
    <=> ( ( A2 = A6 )
        & ( B2 = B6 ) ) ) ).

% old.prod.inject
tff(fact_1441_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X23) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y2 ) ) ) ).

% prod.inject
tff(fact_1442_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_1443_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_1444_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_1445_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_1446_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : finite_finite2(A,aa(list(A),set(A),set2(A),Xs)) ).

% List.finite_set
tff(fact_1447_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% minus_mod_self2
tff(fact_1448_mod__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( modulo_modulo(nat,Mb,Nb) = Mb ) ) ).

% mod_less
tff(fact_1449_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_1450_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_1451_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_1452_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_1453_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_1454_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_1455_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_1456_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_1457_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_1458_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_1459_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_1460_mod__by__Suc__0,axiom,
    ! [Mb: nat] : modulo_modulo(nat,Mb,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_1461_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,Mb: 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),Nb),K)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self4
tff(fact_1462_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,Mb: 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),Nb)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self3
tff(fact_1463_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self2
tff(fact_1464_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self1
tff(fact_1465_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% one_mod_two_eq_one
tff(fact_1466_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_1467_mod2__Suc__Suc,axiom,
    ! [Mb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,Mb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% mod2_Suc_Suc
tff(fact_1468_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: 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)),Nb)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1469_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),aa(num,num,bit0,one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1470_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),aa(num,num,bit0,one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1471_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1472_add__self__mod__2,axiom,
    ! [Mb: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_1473_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),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))),Mb),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_1474_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : modulo_modulo(nat,Mb,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = modulo_modulo(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_1475_mod2__gr__0,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
    <=> ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_1476_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_1477_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_1478_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_1479_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_1480_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A6,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),A6),B6),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_1481_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_1482_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_1483_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_1484_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A6,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),A6),B6),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_1485_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_1486_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) ) ).

% mod_diff_right_eq
tff(fact_1487_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) ) ).

% mod_diff_left_eq
tff(fact_1488_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A6,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B6,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A6),B6),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_1489_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C2) ) ).

% mod_diff_eq
tff(fact_1490_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,Nb: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),modulo_modulo(A,A2,B2)),Nb),B2) = modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),B2) ) ).

% power_mod
tff(fact_1491_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,modulo_modulo(nat,Mb,Nb))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,Mb)),Nb) ).

% mod_Suc_Suc_eq
tff(fact_1492_mod__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,modulo_modulo(nat,Mb,Nb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% mod_Suc_eq
tff(fact_1493_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,Mb,Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_1494_bounded__nat__set__is__finite,axiom,
    ! [N2: set(nat),Nb: nat] :
      ( ! [X2: nat] :
          ( aa(set(nat),$o,member(nat,X2),N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),Nb) )
     => finite_finite2(nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_1495_finite__nat__set__iff__bounded,axiom,
    ! [N2: set(nat)] :
      ( finite_finite2(nat,N2)
    <=> ? [M5: nat] :
        ! [X: nat] :
          ( aa(set(nat),$o,member(nat,X),N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1496_finite__nat__set__iff__bounded__le,axiom,
    ! [N2: set(nat)] :
      ( finite_finite2(nat,N2)
    <=> ? [M5: nat] :
        ! [X: nat] :
          ( aa(set(nat),$o,member(nat,X),N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1497_finite__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A3 ) ).

% finite_list
tff(fact_1498_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I2: nat] : finite_finite2(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(fun(nat,$o),fun(nat,fun(nat,$o)),P),I2))) ).

% finite_M_bounded_by_nat
tff(fact_1499_finite__less__ub,axiom,
    ! [F2: fun(nat,nat),U: nat] :
      ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,F2,N))
     => finite_finite2(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(fun(nat,nat),fun(nat,fun(nat,$o)),F2),U))) ) ).

% finite_less_ub
tff(fact_1500_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( finite_finite2(A,A3)
     => finite_finite2(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_aj(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_eq
tff(fact_1501_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),modulo_modulo(A,A2,B2)),A2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1502_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A2,B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1503_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1504_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_1505_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),one2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_1506_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D3: 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),D3)) ) ) ).

% mod_eqE
tff(fact_1507_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_1508_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) = $ite(aa(nat,nat,suc,modulo_modulo(nat,Mb,Nb)) = Nb,zero_zero(nat),aa(nat,nat,suc,modulo_modulo(nat,Mb,Nb))) ).

% mod_Suc
tff(fact_1509_mod__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat,P2: nat,Mb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),P2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),P2)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),P2)
               => ( aa(nat,$o,P,N)
                 => aa(nat,$o,P,modulo_modulo(nat,aa(nat,nat,suc,N),P2)) ) )
           => aa(nat,$o,P,Mb) ) ) ) ) ).

% mod_induct
tff(fact_1510_mod__less__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),modulo_modulo(nat,Mb,Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_1511_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,Mb,aa(nat,nat,suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_1512_mod__eq__0D,axiom,
    ! [Mb: nat,D2: nat] :
      ( ( modulo_modulo(nat,Mb,D2) = zero_zero(nat) )
     => ? [Q3: nat] : Mb = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q3) ) ).

% mod_eq_0D
tff(fact_1513_mod__geq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( modulo_modulo(nat,Mb,Nb) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb) ) ) ).

% mod_geq
tff(fact_1514_mod__if,axiom,
    ! [Mb: nat,Nb: nat] :
      modulo_modulo(nat,Mb,Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb),Mb,modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ).

% mod_if
tff(fact_1515_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( modulo_modulo(nat,Mb,Nb) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb) ) ) ).

% le_mod_geq
tff(fact_1516_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xb,Nb) = modulo_modulo(nat,Y,Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1517_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_1518_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( finite_finite2(A,A3)
     => finite_finite2(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ak(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_le
tff(fact_1519_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( modulo_modulo(A,A2,B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1520_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A2,B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1521_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1522_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_1523_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(6)
tff(fact_1524_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(8)
tff(fact_1525_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_1526_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_1527_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_1528_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_1529_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_1530_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_1531_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_1532_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_1533_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_1534_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_1535_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_1536_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_1537_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(10)
tff(fact_1538_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) ) ).

% cong_exp_iff_simps(12)
tff(fact_1539_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_1540_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,Mb,Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_1541_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Mb,Q2) = modulo_modulo(nat,Nb,Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
       => ~ ! [S2: nat] : Mb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_1542_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Mb,Q2) = modulo_modulo(nat,Nb,Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ~ ! [S2: nat] : Nb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_1543_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( modulo_modulo(nat,Xb,Nb) = modulo_modulo(nat,Y,Nb) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xb)
       => ? [Q3: nat] : Xb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1544_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_1545_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : modulo_modulo(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb),Q2))),modulo_modulo(nat,Mb,Nb)) ).

% mod_mult2_eq
tff(fact_1546_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique8689654367752047608divmod(nat,Mb,Nb) = 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),Mb)),aa(num,nat,numeral_numeral(nat),Nb))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),Mb),aa(num,nat,numeral_numeral(nat),Nb))) ).

% divmod'_nat_def
tff(fact_1547_modulo__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : modulo_modulo(nat,Mb,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_1548_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6) )
     => ~ ( ( A2 = A6 )
         => ( B2 != B6 ) ) ) ).

% Pair_inject
tff(fact_1549_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o),P2: product_prod(A,B)] :
      ( ! [A4: A,B3: B] : aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
     => aa(product_prod(A,B),$o,P,P2) ) ).

% prod_cases
tff(fact_1550_surj__pair,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B)] :
    ? [X2: A,Y4: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y4) ).

% surj_pair
tff(fact_1551_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A4: A,B3: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) ).

% old.prod.exhaust
tff(fact_1552_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_1553_split__mod,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,modulo_modulo(nat,Mb,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,Mb) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I4)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_1554_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : unique8689654367752047608divmod(A,Mb,Nb) = 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),Mb)),aa(num,A,numeral_numeral(A),Nb))),modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Nb))) ) ).

% divmod_def
tff(fact_1555_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1556_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_1557_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Mb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_1558_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),Nb) = 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))),Mb),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_1559_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Mb)
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),Mb) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1560_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),B2)) = modulo_modulo(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1561_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),aa(num,num,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),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_1562_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat,Mb: 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),aa(num,num,bit0,one2))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)) = 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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) ) ).

% div_exp_mod_exp_eq
tff(fact_1563_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1564_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( 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),aa(num,num,bit0,one2))),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = 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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1565_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,C)),$o),Xb: product_prod(A,product_prod(B,C))] :
      ( ! [A4: A,B3: B,C3: C] : aa(product_prod(A,product_prod(B,C)),$o,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)))
     => aa(product_prod(A,product_prod(B,C)),$o,P,Xb) ) ).

% prod_induct3
tff(fact_1566_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A4: A,B3: B,C3: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)) ).

% prod_cases3
tff(fact_1567_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: 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),R))
    <=> ( ( 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)),R) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int)) ),
              Q2 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1568_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( ( modulo_modulo(A,Xb,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)) = modulo_modulo(A,Xb,Mb) )
              | ( modulo_modulo(A,Xb,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,Xb,Mb)),Mb) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_1569_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),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),aa(num,num,bit0,one2))),B2)))
             => ( aa(A,A,aa(A,fun(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),aa(num,num,bit0,one2))),B2))),B2) = modulo_modulo(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1570_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% unset_bit_Suc
tff(fact_1571_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% set_bit_Suc
tff(fact_1572_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),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),aa(num,num,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),aa(num,num,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),aa(num,num,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_1573_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),R))
       => 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),aa(num,num,bit0,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_1574_prod__induct7,axiom,
    ! [G2: $tType,F: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))))] :
      ( ! [A4: A,B3: B,C3: C,D3: D,E2: E3,F3: F,G3: G2] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),B3),aa(product_prod(D,product_prod(E3,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),aa(C,fun(product_prod(D,product_prod(E3,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_Pair(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),C3),aa(product_prod(E3,product_prod(F,G2)),product_prod(D,product_prod(E3,product_prod(F,G2))),aa(D,fun(product_prod(E3,product_prod(F,G2)),product_prod(D,product_prod(E3,product_prod(F,G2)))),product_Pair(D,product_prod(E3,product_prod(F,G2))),D3),aa(product_prod(F,G2),product_prod(E3,product_prod(F,G2)),aa(E3,fun(product_prod(F,G2),product_prod(E3,product_prod(F,G2))),product_Pair(E3,product_prod(F,G2)),E2),aa(G2,product_prod(F,G2),aa(F,fun(G2,product_prod(F,G2)),product_Pair(F,G2),F3),G3)))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),$o,P,Xb) ) ).

% prod_induct7
tff(fact_1575_prod__induct6,axiom,
    ! [F: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))))] :
      ( ! [A4: A,B3: B,C3: C,D3: D,E2: E3,F3: F] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),A4),aa(product_prod(C,product_prod(D,product_prod(E3,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E3,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E3,F)))),B3),aa(product_prod(D,product_prod(E3,F)),product_prod(C,product_prod(D,product_prod(E3,F))),aa(C,fun(product_prod(D,product_prod(E3,F)),product_prod(C,product_prod(D,product_prod(E3,F)))),product_Pair(C,product_prod(D,product_prod(E3,F))),C3),aa(product_prod(E3,F),product_prod(D,product_prod(E3,F)),aa(D,fun(product_prod(E3,F),product_prod(D,product_prod(E3,F))),product_Pair(D,product_prod(E3,F)),D3),aa(F,product_prod(E3,F),aa(E3,fun(F,product_prod(E3,F)),product_Pair(E3,F),E2),F3))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,F))))),$o,P,Xb) ) ).

% prod_induct6
tff(fact_1576_prod__induct5,axiom,
    ! [E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3))))] :
      ( ! [A4: A,B3: B,C3: C,D3: D,E2: E3] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,E3))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E3))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E3)))),A4),aa(product_prod(C,product_prod(D,E3)),product_prod(B,product_prod(C,product_prod(D,E3))),aa(B,fun(product_prod(C,product_prod(D,E3)),product_prod(B,product_prod(C,product_prod(D,E3)))),product_Pair(B,product_prod(C,product_prod(D,E3))),B3),aa(product_prod(D,E3),product_prod(C,product_prod(D,E3)),aa(C,fun(product_prod(D,E3),product_prod(C,product_prod(D,E3))),product_Pair(C,product_prod(D,E3)),C3),aa(E3,product_prod(D,E3),aa(D,fun(E3,product_prod(D,E3)),product_Pair(D,E3),D3),E2)))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E3)))),$o,P,Xb) ) ).

% prod_induct5
tff(fact_1577_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,D))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ( ! [A4: A,B3: B,C3: C,D3: D] : aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D3))))
     => aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,Xb) ) ).

% prod_induct4
tff(fact_1578_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))))] :
      ~ ! [A4: A,B3: B,C3: C,D3: D,E2: E3,F3: F,G3: G2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),B3),aa(product_prod(D,product_prod(E3,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),aa(C,fun(product_prod(D,product_prod(E3,product_prod(F,G2))),product_prod(C,product_prod(D,product_prod(E3,product_prod(F,G2))))),product_Pair(C,product_prod(D,product_prod(E3,product_prod(F,G2)))),C3),aa(product_prod(E3,product_prod(F,G2)),product_prod(D,product_prod(E3,product_prod(F,G2))),aa(D,fun(product_prod(E3,product_prod(F,G2)),product_prod(D,product_prod(E3,product_prod(F,G2)))),product_Pair(D,product_prod(E3,product_prod(F,G2))),D3),aa(product_prod(F,G2),product_prod(E3,product_prod(F,G2)),aa(E3,fun(product_prod(F,G2),product_prod(E3,product_prod(F,G2))),product_Pair(E3,product_prod(F,G2)),E2),aa(G2,product_prod(F,G2),aa(F,fun(G2,product_prod(F,G2)),product_Pair(F,G2),F3),G3)))))) ).

% prod_cases7
tff(fact_1579_verit__le__mono__div,axiom,
    ! [A3: nat,B4: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),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),Nb)),
                $ite(modulo_modulo(nat,B4,Nb) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_1580_mod__exhaust__less__4,axiom,
    ! [Mb: nat] :
      ( ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
      | ( modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_1581_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [A3: fun(B,A),B4: fun(B,A),Xb: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),minus_minus(fun(B,A)),A3),B4),Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A3,Xb)),aa(B,A,B4,Xb)) ) ).

% minus_apply
tff(fact_1582_finite__Collect__le__nat,axiom,
    ! [K: nat] : finite_finite2(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_al(nat,fun(nat,$o)),K))) ).

% finite_Collect_le_nat
tff(fact_1583_finite__Collect__less__nat,axiom,
    ! [K: nat] : finite_finite2(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),K))) ).

% finite_Collect_less_nat
tff(fact_1584_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => finite_finite2(set(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_an(set(A),fun(set(A),$o),A3))) ) ).

% finite_Collect_subsets
tff(fact_1585_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ao(nat,fun(A,$o),Nb))) ) ) ).

% finite_roots_unity
tff(fact_1586_finite__Diff2,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( finite_finite2(A,B4)
     => ( finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))
      <=> finite_finite2(A,A3) ) ) ).

% finite_Diff2
tff(fact_1587_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y2: num] :
      ( ( aa(num,num,bit0,X23) = aa(num,num,bit0,Y2) )
    <=> ( X23 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_1588_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa(num,num,bit1,X32) = aa(num,num,bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_1589_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),P))
        | finite_finite2(A,aa(fun(A,$o),set(A),collect(A),Q)) )
     => finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_1590_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aq(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),P))
        & finite_finite2(A,aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% finite_Collect_disjI
tff(fact_1591_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ar(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_1592_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_as(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_1593_finite__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( finite_finite2(A,A3)
     => finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) ) ).

% finite_Diff
tff(fact_1594_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_at(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_1595_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_au(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_1596_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1597_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1598_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,V)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W))) ).

% zmod_numeral_Bit0
tff(fact_1599_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),aa(num,num,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),aa(num,num,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_1600_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,K,L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1601_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),modulo_modulo(int,K,L)) ) ).

% neg_mod_bound
tff(fact_1602_zmod__eq__0__iff,axiom,
    ! [Mb: int,D2: int] :
      ( ( modulo_modulo(int,Mb,D2) = zero_zero(int) )
    <=> ? [Q4: int] : Mb = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q4) ) ).

% zmod_eq_0_iff
tff(fact_1603_zmod__eq__0D,axiom,
    ! [Mb: int,D2: int] :
      ( ( modulo_modulo(int,Mb,D2) = zero_zero(int) )
     => ? [Q3: int] : Mb = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q3) ) ).

% zmod_eq_0D
tff(fact_1604_finite__maxlen,axiom,
    ! [A: $tType,M6: set(list(A))] :
      ( finite_finite2(list(A),M6)
     => ? [N: nat] :
        ! [X4: list(A)] :
          ( aa(set(list(A)),$o,member(list(A),X4),M6)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X4)),N) ) ) ).

% finite_maxlen
tff(fact_1605_div__mod__decomp__int,axiom,
    ! [A3: int,Nb: 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),Nb)),Nb)),modulo_modulo(int,A3,Nb)) ).

% div_mod_decomp_int
tff(fact_1606_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,K,L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_1607_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1608_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% verit_la_disequality
tff(fact_1609_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% verit_comp_simplify1(2)
tff(fact_1610_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% verit_comp_simplify1(1)
tff(fact_1611_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),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_1612_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1613_verit__le__mono__div__int,axiom,
    ! [A3: int,B4: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
       => aa(int,$o,
            aa(int,fun(int,$o),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),Nb)),
                $ite(modulo_modulo(int,B4,Nb) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_1614_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ finite_finite2(A,aa(fun(A,$o),set(A),collect(A),P))
     => ? [X_13: A] : aa(A,$o,P,X_13) ) ).

% not_finite_existsD
tff(fact_1615_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),R2: fun(A,fun(B,$o))] :
      ( ~ finite_finite2(A,A3)
     => ( finite_finite2(B,B4)
       => ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => ? [Xa: B] :
                  ( aa(set(B),$o,member(B,Xa),B4)
                  & aa(B,$o,aa(A,fun(B,$o),R2,X2),Xa) ) )
         => ? [X2: B] :
              ( aa(set(B),$o,member(B,X2),B4)
              & ~ finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_av(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A3),R2),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_1616_split__zmod,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,modulo_modulo(int,Nb,K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,Nb) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_1617_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: 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)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R)
         => ( modulo_modulo(int,A2,B2) = R ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1618_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R: 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)),R) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R),B2)
         => ( modulo_modulo(int,A2,B2) = R ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1619_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_1620_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K)),modulo_modulo(int,Nb,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1621_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K)),modulo_modulo(int,Nb,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1622_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B6: A,A6: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B6),A6)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A6),B6) ) ) ).

% verit_comp_simplify1(3)
tff(fact_1623_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_1624_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one2 != aa(num,num,bit0,X23) ).

% verit_eq_simplify(10)
tff(fact_1625_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),A2)
                & ! [Xa: A] :
                    ( aa(set(A),$o,member(A,Xa),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1626_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X2)
                & ! [Xa: A] :
                    ( aa(set(A),$o,member(A,Xa),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1627_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X32: num] : aa(num,num,bit0,X23) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_1628_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_1629_Diff__infinite__finite,axiom,
    ! [A: $tType,T2: set(A),S: set(A)] :
      ( finite_finite2(A,T2)
     => ( ~ finite_finite2(A,S)
       => ~ finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) ) ) ).

% Diff_infinite_finite
tff(fact_1630_eq__diff__eq_H,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( ( Xb = aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),Z) )
    <=> ( Y = aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Z) ) ) ).

% eq_diff_eq'
tff(fact_1631_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),modulo_modulo(int,B2,A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_1632_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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_1633_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A3: fun(A,B),B4: fun(A,B),X4: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A3),B4),X4) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,X4)),aa(A,B,B4,X4)) ) ).

% fun_diff_def
tff(fact_1634_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
                & ! [Xa: A] :
                    ( aa(set(A),$o,member(A,Xa),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1635_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
                & ! [Xa: A] :
                    ( aa(set(A),$o,member(A,Xa),A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1636_div__less__mono,axiom,
    ! [A3: nat,B4: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( modulo_modulo(nat,A3,Nb) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B4,Nb) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_1637_div__mod__decomp,axiom,
    ! [A3: nat,Nb: 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),Nb)),Nb)),modulo_modulo(nat,A3,Nb)) ).

% div_mod_decomp
tff(fact_1638_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1639_finite__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => finite_finite2(complex,aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_aw(nat,fun(complex,fun(complex,$o)),Nb),C2))) ) ).

% finite_nth_roots
tff(fact_1640_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se8732182000553998342ip_bit(A,Nb,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% flip_bit_Suc
tff(fact_1641_product__nth,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),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)),Nb) = 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),Nb),aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,Nb,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_1642_old_Oprod_Orec,axiom,
    ! [B: $tType,A: $tType,C: $tType,F1: fun(B,fun(C,A)),A2: B,B2: C] : product_rec_prod(B,C,A,F1,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),F1,A2),B2) ).

% old.prod.rec
tff(fact_1643_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ay(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_1644_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ba(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_1645_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_1646_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_1647_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_1648_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_1649_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_1650_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_1651_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : neg_numeral_dbl_inc(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_1652_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Leaf((A2),(B2))) = zero_zero(nat) ).

% VEBT_internal.height.simps(1)
tff(fact_1653_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Mb: nat,Nb: nat] :
      ( ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M),zero_zero(nat))
     => ( ! [M: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N),modulo_modulo(nat,M,N))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M),N) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_1654_insert__simp__norm,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),Xb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
         => ( ( Xb != Maa )
           => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = 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),Mia),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xb),Maa))),Dega,list_update(vEBT_VEBT,TreeLista,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_1655_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list(vEBT_VEBT),Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega)
         => ( ( Xb != Maa )
           => ( 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),Mia),Maa)),Dega,TreeLista,Summarya),Xb) = 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),Xb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mia),Maa))),Dega,list_update(vEBT_VEBT,TreeLista,vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_1656_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,Nb),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,bit_concat_bit(Nb,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),L))) ).

% concat_bit_Suc
tff(fact_1657_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).

% dbl_simps(3)
tff(fact_1658_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.bounded_iff
tff(fact_1659_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_1660_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_1661_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb3
tff(fact_1662_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_1663_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_1664_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,Xb: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,I2,Xb)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_1665_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),Nb)) ).

% max_Suc_Suc
tff(fact_1666_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(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_1667_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_1668_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1669_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_1670_max__0L,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),Nb) = Nb ).

% max_0L
tff(fact_1671_max__0R,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),zero_zero(nat)) = Nb ).

% max_0R
tff(fact_1672_list__update__id,axiom,
    ! [A: $tType,Xs: list(A),I2: nat] : list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),I2)) = Xs ).

% list_update_id
tff(fact_1673_nth__list__update__neq,axiom,
    ! [A: $tType,I2: nat,J: nat,Xs: list(A),Xb: A] :
      ( ( I2 != J )
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,Xb)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_1674_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_1675_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_1676_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_1677_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(3)
tff(fact_1678_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xb)),zero_zero(A)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(4)
tff(fact_1679_max__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),one_one(A)) = one_one(A) ) ) ).

% max_0_1(1)
tff(fact_1680_max__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),zero_zero(A)) = one_one(A) ) ) ).

% max_0_1(2)
tff(fact_1681_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(6)
tff(fact_1682_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(5)
tff(fact_1683_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2)
     => ( list_update(A,Xs,I2,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_1684_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_concat_bit(Nb,K),L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_1685_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_concat_bit(Nb,K),L)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_1686_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),aa(num,num,bit0,K)) ) ).

% dbl_simps(5)
tff(fact_1687_nth__list__update__eq,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,Xb)),I2) = Xb ) ) ).

% nth_list_update_eq
tff(fact_1688_set__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I2))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_1689_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ) ).

% max.mono
tff(fact_1690_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.orderE
tff(fact_1691_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% max.orderI
tff(fact_1692_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.boundedE
tff(fact_1693_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2) ) ) ) ).

% max.boundedI
tff(fact_1694_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.order_iff
tff(fact_1695_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded1
tff(fact_1696_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded2
tff(fact_1697_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_1698_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb_iff1
tff(fact_1699_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_1700_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI1
tff(fact_1701_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI2
tff(fact_1702_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_1703_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% max.strict_boundedE
tff(fact_1704_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_1705_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_1706_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_1707_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_1708_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ).

% max_add_distrib_right
tff(fact_1709_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_1710_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),Nb)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Q2)) ).

% nat_add_max_left
tff(fact_1711_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q2)) ).

% nat_add_max_right
tff(fact_1712_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)) ).

% nat_mult_max_right
tff(fact_1713_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),Nb)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ).

% nat_mult_max_left
tff(fact_1714_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: A,Xa: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X4),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa),Xa,X4) ) ).

% max_def_raw
tff(fact_1715_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,Mb: nat,L: int,R: int] : aa(int,int,bit_concat_bit(Nb,K),aa(int,int,bit_concat_bit(Mb,L),R)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb),aa(int,int,bit_concat_bit(Nb,K),L)),R) ).

% concat_bit_assoc
tff(fact_1716_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),Mb) ).

% nat_minus_add_max
tff(fact_1717_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),Xb: A,I2: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
     => ( aa(set(A),$o,member(A,Xb),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,Xb))),A3) ) ) ).

% set_update_subsetI
tff(fact_1718_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : neg_numeral_dbl(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb) ) ).

% dbl_def
tff(fact_1719_set__update__memI,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_1720_list__update__same__conv,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I2,Xb) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I2) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_1721_nth__list__update,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Xb: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,Xb)),J) = $ite(I2 = J,Xb,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_1722_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A4: nat,B3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B3),A4) )
     => ( ! [A4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A4),zero_zero(nat))
       => ( ! [A4: nat,B3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_1723_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] : Xb != vEBT_Leaf((A4),(B3))
     => ~ ! [Uu2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb != vEBT_Node(Uu2,Deg,TreeList,Summary) ) ).

% VEBT_internal.height.cases
tff(fact_1724_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,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),aa(num,num,bit0,one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            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),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Mia),Xb,Mia)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Maa))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeLista,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(xn,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),aa(num,num,bit0,one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_1725_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B3))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
              ( ( Xb = 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] :
                ( ( Xb = 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) ) )
           => ( ! [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary) )
                 => ( 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),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = 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,Va3)),TreeList,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = 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),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Xaa,Mi)),
                                    aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),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,Va3)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_1726_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = 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),Xb),Xaa))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),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))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = 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)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = 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)),Xaa)) ) )
             => ( ! [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary) )
                   => ( ( 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),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary) )
                     => ~ 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,V2)),TreeList,Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = 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,Va3)),TreeList,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = 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),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi),Xaa,Mi)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),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,Va3)),TreeList,Summary) ) ) ) )
                       => ~ 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_1727_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( 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),aa(num,num,bit0,one2))),Nb)) = 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),aa(num,num,bit0,one2))),Nb))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_1728_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Nb: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ? [Y5: A] :
                ( aa(A,$o,P,Y5)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y5)),aa(A,nat,F2,X2)) ) )
       => ? [Y4: A] :
            ( aa(A,$o,P,Y4)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),Nb)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1729_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,Mb),aa(num,num,bit0,Nb)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_bb(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Mb,Nb)) ) ).

% divmod_algorithm_code(6)
tff(fact_1730_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( 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),aa(num,num,bit0,one2))),Nb)) = 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),aa(num,num,bit0,one2))),Nb)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_1731_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X23: A] : aa(option(A),nat,size_option(A,Xb),aa(A,option(A),some(A),X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_1732_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_1733_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),one_one(nat))
    <=> ( Mb = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_1734_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Mb = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_1735_dvd__1__left,axiom,
    ! [K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K) ).

% dvd_1_left
tff(fact_1736_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1737_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1738_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ) ).

% div_dvd_div
tff(fact_1739_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( ( K = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1740_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_1741_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,aa(fun(B,fun(C,A)),fun(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_1742_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1743_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1744_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1745_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1746_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_1747_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1748_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1749_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1750_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1751_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1752_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => aa(A,$o,aa(A,fun(A,$o),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_1753_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_1754_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),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_1755_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(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_1756_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Nb)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_1757_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_1758_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Nb)))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_1759_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,Nb))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) ) ).

% even_Suc
tff(fact_1760_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1761_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1762_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_1763_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2) ) ) ) ).

% even_mult_iff
tff(fact_1764_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2) ) ) ) ).

% even_add
tff(fact_1765_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ~ ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
            <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2) ) ) ) ).

% odd_add
tff(fact_1766_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ).

% even_mod_2_iff
tff(fact_1767_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ).

% even_Suc_div_two
tff(fact_1768_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_1769_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_1770_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,Nb,Mb)) ) ) ).

% dvd_numeral_simp
tff(fact_1771_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1772_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1773_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),zero_zero(A))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_1774_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ).

% even_plus_one_iff
tff(fact_1775_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ).

% even_diff
tff(fact_1776_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_1777_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_1778_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
              & ( A2 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1779_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1780_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,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),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1781_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1782_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% even_power
tff(fact_1783_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_1784_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,Mb),aa(num,num,bit0,Nb)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_bc(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Mb,Nb)) ) ).

% divmod_algorithm_code(5)
tff(fact_1785_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1786_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W))
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),W))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1787_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)),one_one(A)))
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1788_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_1789_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,H: fun(B,A),F2: fun(C,fun(D,B)),Prod: product_prod(C,D)] : aa(B,A,H,aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),F2),Prod)) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_bd(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),H),F2)),Prod) ).

% prod.case_distrib
tff(fact_1790_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ~ ! [K2: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvdE
tff(fact_1791_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A2: A,B2: A,K: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% dvdI
tff(fact_1792_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
        <=> ? [K3: A] : A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvd_def
tff(fact_1793_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult
tff(fact_1794_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult2
tff(fact_1795_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ).

% dvd_mult_left
tff(fact_1796_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% dvd_triv_left
tff(fact_1797_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ).

% mult_dvd_mono
tff(fact_1798_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ).

% dvd_mult_right
tff(fact_1799_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) ) ).

% dvd_triv_right
tff(fact_1800_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ~ ! [X2: A,Y4: A] :
                ( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y4) )
               => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),X2),A2)
                 => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Y4),B2) ) ) ) ) ).

% dvd_productE
tff(fact_1801_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ? [B7: A,C6: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B7),C6) )
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B7),B2)
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C6),C2) ) ) ) ).

% division_decomp
tff(fact_1802_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),one_one(A)),A2) ) ).

% one_dvd
tff(fact_1803_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% unit_imp_dvd
tff(fact_1804_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1805_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_1806_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_add_left_iff
tff(fact_1807_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ) ).

% dvd_add
tff(fact_1808_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) ) ) ).

% dvd_diff_commute
tff(fact_1809_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Z)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ) ) ).

% dvd_diff
tff(fact_1810_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),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),D2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% div_div_div_same
tff(fact_1811_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1812_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),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_1813_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) ) ) ).

% dvd_power_same
tff(fact_1814_old_Oprod_Ocase,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,fun(C,A)),X1: B,X23: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(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),X1),X23)) = aa(C,A,aa(B,fun(C,A),F2,X1),X23) ).

% old.prod.case
tff(fact_1815_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( modulo_modulo(A,modulo_modulo(A,A2,B2),C2) = modulo_modulo(A,A2,C2) ) ) ) ).

% mod_mod_cancel
tff(fact_1816_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K: A,Mb: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),K),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),K),Nb)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),K),modulo_modulo(A,Mb,Nb)) ) ) ) ).

% dvd_mod
tff(fact_1817_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% signed_take_bit_mult
tff(fact_1818_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_1819_dvd__diff__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_1820_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% signed_take_bit_diff
tff(fact_1821_zdvd__zdiffD,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),Nb))
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),Nb)
       => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),Mb) ) ) ).

% zdvd_zdiffD
tff(fact_1822_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(A,fun(A,$o),B2)))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% subset_divisors_dvd
tff(fact_1823_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,$o),P: fun(B,fun(C,A)),Z: product_prod(B,C)] :
      ( aa(A,$o,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z))
     => ~ ! [X2: B,Y4: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y4) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X2),Y4)) ) ) ).

% case_prodE2
tff(fact_1824_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_bf(fun(product_prod(A,B),C),fun(A,fun(B,C)),F2)) = F2 ).

% case_prod_eta
tff(fact_1825_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X2: A,Y4: B] : aa(B,C,aa(A,fun(B,C),F2,X2),Y4) = 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),X2),Y4))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2) = G ) ) ).

% cond_case_prod_eta
tff(fact_1826_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_be(A,fun(A,$o),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1827_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,Mb),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ).

% even_signed_take_bit_iff
tff(fact_1828_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),one_one(A)) ) ).

% not_is_unit_0
tff(fact_1829_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S3)) ) ) ) ).

% pinf(9)
tff(fact_1830_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S3))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S3)) ) ) ) ).

% pinf(10)
tff(fact_1831_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S3)) ) ) ) ).

% minf(9)
tff(fact_1832_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z4: A] :
        ! [X4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S3))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),S3)) ) ) ) ).

% minf(10)
tff(fact_1833_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1834_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_1835_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_1836_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_1837_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1838_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1839_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1840_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1841_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),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),D2)) = 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),D2)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1842_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_1843_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1844_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),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_1845_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1846_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1847_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1848_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1849_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1850_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1851_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1852_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),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)),Nb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% div_power
tff(fact_1853_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A,Nb: nat,Mb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Mb)) ) ) ) ).

% dvd_power_le
tff(fact_1854_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat,B2: A,Mb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb)),B2) ) ) ) ).

% power_le_dvd
tff(fact_1855_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Mb: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% le_imp_power_dvd
tff(fact_1856_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ).

% mod_eq_dvd_iff
tff(fact_1857_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,A2,B2))) ) ).

% dvd_minus_mod
tff(fact_1858_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
       => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_1859_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb) ) ) ).

% dvd_pos_nat
tff(fact_1860_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_1861_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_1862_dvd__diffD1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Mb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_1863_dvd__diffD,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Mb) ) ) ) ).

% dvd_diffD
tff(fact_1864_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),A2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),B2)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D2) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D2) ) )
         => ? [X2: nat,Y4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),A2)
              & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),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),X2) = 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)),Y4)),D2) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1865_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D3: nat,X2: nat,Y4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),A2)
      & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D3) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_1866_zdvd__mono,axiom,
    ! [K: int,Mb: int,Ta: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Mb),Ta)
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Mb)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Ta)) ) ) ).

% zdvd_mono
tff(fact_1867_zdvd__mult__cancel,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Mb)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Nb))
     => ( ( K != zero_zero(int) )
       => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Mb),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_1868_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D3: nat,X2: nat,Y4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),A2)
      & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)) = D3 )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_1869_zdvd__period,axiom,
    ! [A2: int,D2: int,Xb: int,Ta: int,C2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),D2)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Ta))
      <=> aa(int,$o,aa(int,fun(int,$o),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),Xb),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D2))),Ta)) ) ) ).

% zdvd_period
tff(fact_1870_zdvd__reduce,axiom,
    ! [K: int,Nb: int,Mb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),Mb)))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),K),Nb) ) ).

% zdvd_reduce
tff(fact_1871_finite__divisors__int,axiom,
    ! [I2: int] :
      ( ( I2 != zero_zero(int) )
     => finite_finite2(int,aa(fun(int,$o),set(int),collect(int),aTP_Lamp_bg(int,fun(int,$o),I2))) ) ).

% finite_divisors_int
tff(fact_1872_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xb)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Y) )
       => ( Xb = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_1873_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1874_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X))
        <=> ? [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),zero_zero(A)))
              & aa(A,$o,P,X) ) ) ) ).

% unity_coeff_ex
tff(fact_1875_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),D2)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D2),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),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1876_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1877_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2)
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1878_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),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_1879_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1880_even__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb))) ) ).

% even_numeral
tff(fact_1881_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D5: A,Ta: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),D5)
         => ! [X4: A,K4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Ta))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),Ta)) ) ) ) ).

% inf_period(3)
tff(fact_1882_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D5: A,Ta: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),D5)
         => ! [X4: A,K4: A] :
              ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Ta))
            <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),Ta)) ) ) ) ).

% inf_period(4)
tff(fact_1883_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1884_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1885_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),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_1886_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1887_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_1888_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),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_1889_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1890_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1891_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% dvd_imp_le
tff(fact_1892_dvd__mult__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_1893_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1894_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D3: nat,X2: nat,Y4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),A2)
          & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),B2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1895_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,Mb,Nb))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1896_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( ( modulo_modulo(nat,Mb,Q2) = modulo_modulo(nat,Nb,Q2) )
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1897_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,Mb: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ? [X2: A] :
          ( aa(A,$o,P,X2)
          & ! [Y5: A] :
              ( aa(A,$o,P,Y5)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Mb,X2)),aa(A,nat,Mb,Y5)) ) ) ) ).

% ex_has_least_nat
tff(fact_1898_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod(nat,nat)] :
      ~ ! [K2: nat,M: nat] : Xb != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K2),M) ).

% prod_decode_aux.cases
tff(fact_1899_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => finite_finite2(nat,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bh(nat,fun(nat,$o),Mb))) ) ).

% finite_divisors_nat
tff(fact_1900_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),zero_zero(A)) ) ).

% even_zero
tff(fact_1901_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B3: A] :
                  ( ( B3 != zero_zero(A) )
                 => ( aa(A,$o,aa(A,fun(A,$o),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_1902_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),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_1903_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),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_1904_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
         => ~ ! [B3: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B3) ) ) ).

% evenE
tff(fact_1905_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),one_one(A)) ) ).

% odd_one
tff(fact_1906_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% odd_even_add
tff(fact_1907_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2) )
            & ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_1908_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xb: A,Mb: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb))
          <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ) ).

% dvd_power_iff
tff(fact_1909_odd__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb))) ) ).

% odd_numeral
tff(fact_1910_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat,Xb: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
            | ( Xb = one_one(A) ) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xb),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) ) ) ).

% dvd_power
tff(fact_1911_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2))))) ) ).

% even_even_mod_4_iff
tff(fact_1912_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),Mb)
      <=> ( Nb = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_1913_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Mb)),Mb)
      <=> ( Nb = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_1914_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),Mb))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Q2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),Mb)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_1915_power__dvd__imp__le,axiom,
    ! [I2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I2),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_1916_mod__nat__eqI,axiom,
    ! [R: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),Mb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),R))
         => ( modulo_modulo(nat,Mb,Nb) = R ) ) ) ) ).

% mod_nat_eqI
tff(fact_1917_aset_I10_J,axiom,
    ! [D2: int,D5: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A3)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),Ta))
           => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5)),Ta)) ) ) ) ).

% aset(10)
tff(fact_1918_aset_I9_J,axiom,
    ! [D2: int,D5: int,A3: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),A3)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),Ta))
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D5)),Ta)) ) ) ) ).

% aset(9)
tff(fact_1919_bset_I10_J,axiom,
    ! [D2: int,D5: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),Ta))
           => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5)),Ta)) ) ) ) ).

% bset(10)
tff(fact_1920_bset_I9_J,axiom,
    ! [D2: int,D5: int,B4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),D5)
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( aa(set(int),$o,member(int,Xa3),set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( aa(set(int),$o,member(int,Xb3),B4)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),Ta))
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D5)),Ta)) ) ) ) ).

% bset(9)
tff(fact_1921_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_1922_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1923_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1924_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono_odd
tff(fact_1925_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% odd_pos
tff(fact_1926_dvd__power__iff__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_1927_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_1928_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Mb),A2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
            | ( Mb = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_1929_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Mb),A2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
            & ( Mb != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_1930_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se8732182000553998342ip_bit(A,Mb,A2))
        <=> ~ ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
            <=> ( Mb = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_1931_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_diff_iff
tff(fact_1932_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),B3)),one_one(A)) ) ) ).

% oddE
tff(fact_1933_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) ) )
         => ~ ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
             => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_1934_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_1935_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% zero_le_even_power
tff(fact_1936_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_1937_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_1938_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_1939_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_1940_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ~ ! [N: nat] : Xb != aa(nat,nat,suc,N) ) ).

% list_decode.cases
tff(fact_1941_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( ( Nb = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
              & ( A2 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_1942_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),B2: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y4: A] :
            ( aa(A,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y4)),B2) )
       => ? [X2: A] :
            ( aa(A,$o,P,X2)
            & ! [Y5: A] :
                ( aa(A,$o,P,Y5)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y5)),aa(A,nat,F2,X2)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_1943_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_1944_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% even_mask_div_iff'
tff(fact_1945_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1946_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,nat)] : aa(option(A),nat,size_option(A,Xb),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_1947_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% even_mod_4_div_2
tff(fact_1948_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),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bi(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_1949_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% even_mask_div_iff
tff(fact_1950_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),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_bj(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_1951_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% odd_mod_4_div_2
tff(fact_1952_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_1953_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),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_bk(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_1954_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S: set(A)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
                & ~ ? [Xa: A] :
                      ( aa(set(A),$o,member(A,Xa),S)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X2) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1955_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X5: set(A)] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => ? [Xa: A] :
                    ( aa(set(A),$o,member(A,Xa),X5)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Xa) ) )
           => ~ finite_finite2(A,X5) ) ) ) ).

% infinite_growing
tff(fact_1956_triangle__def,axiom,
    ! [Nb: nat] : nat_triangle(Nb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% triangle_def
tff(fact_1957_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va3: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va3)) )
               => ( Y != $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_1958_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $ite(
        ( ( Nb = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ),
        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),Mb),
        aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bl(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_1959_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb),aa(int,int,aa(int,fun(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),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_1960_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = $ite(Nb = zero_zero(nat),aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))) ) ).

% signed_take_bit_rec
tff(fact_1961_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),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_1962_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% flip_bit_0
tff(fact_1963_intind,axiom,
    ! [A: $tType,I2: nat,Nb: nat,P: fun(A,$o),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
     => ( aa(A,$o,P,Xb)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Nb,Xb)),I2)) ) ) ).

% intind
tff(fact_1964_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [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) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A4),B3) )
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P2) ) ).

% case_prodI2
tff(fact_1965_case__prodI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F2,A2),B2)
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),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_1966_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => aa(set(C),$o,member(C,Z),aa(B,set(C),aa(A,fun(B,set(C)),C2,A4),B3)) )
     => aa(set(C),$o,member(C,Z),aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P2)) ) ).

% mem_case_prodI2
tff(fact_1967_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( aa(set(A),$o,member(A,Z),aa(C,set(A),aa(B,fun(C,set(A)),C2,A2),B2))
     => aa(set(A),$o,member(A,Z),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(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_1968_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),Xb: 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 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A4),B3),Xb) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P2),Xb) ) ).

% case_prodI2'
tff(fact_1969_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% neg_le_iff_le
tff(fact_1970_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% neg_less_iff_less
tff(fact_1971_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Mb: num,Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) )
        <=> ( Mb = Nb ) ) ) ).

% neg_numeral_eq_iff
tff(fact_1972_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_1973_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_1974_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_1975_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_1976_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_1977_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_1978_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% minus_diff_eq
tff(fact_1979_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_1980_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_1981_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( (P)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_1982_real__add__minus__iff,axiom,
    ! [Xb: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,uminus_uminus(real),A2)) = zero_zero(real) )
    <=> ( Xb = A2 ) ) ).

% real_add_minus_iff
tff(fact_1983_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( ~ (P)
            & (Q) ) ) ) ).

% of_bool_less_iff
tff(fact_1984_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = one_one(A) )
        <=> (P) ) ) ).

% of_bool_eq_1_iff
tff(fact_1985_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$true) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_1986_length__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),nat,size_size(list(A)),replicate(A,Nb,Xb)) = Nb ).

% length_replicate
tff(fact_1987_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_less_eq_nonneg
tff(fact_1988_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_1989_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_le_0_iff_le
tff(fact_1990_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_1991_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_0_iff_less
tff(fact_1992_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_1993_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_pos
tff(fact_1994_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_1995_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_1996_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_1997_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% verit_minus_simplify(3)
tff(fact_1998_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = aa(A,A,uminus_uminus(A),A2) ) ).

% diff_0
tff(fact_1999_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb))) ) ).

% add_neg_numeral_simps(3)
tff(fact_2000_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_2001_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_2002_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_2003_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,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_2004_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_2005_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Xb) ) ).

% divide_minus1
tff(fact_2006_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% minus_mod_self1
tff(fact_2007_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P)))
        <=> (P) ) ) ).

% zero_less_of_bool_iff
tff(fact_2008_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A))
        <=> ~ (P) ) ) ).

% of_bool_less_one_iff
tff(fact_2009_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: $o] : aa($o,A,zero_neq_one_of_bool(A),~ (P)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% of_bool_not_iff
tff(fact_2010_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa($o,nat,zero_neq_one_of_bool(nat),Nb != aa(nat,nat,suc,zero_zero(nat))) ).

% Suc_0_mod_eq
tff(fact_2011_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),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_2012_Ball__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
         => aa(A,$o,P,X) )
    <=> ( aa(A,$o,P,A2)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_2013_Bex__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ? [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
          & aa(A,$o,P,X) )
    <=> ( aa(A,$o,P,A2)
        & ( Nb != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_2014_in__set__replicate,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Y: A] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),replicate(A,Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_2015_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2016_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,Nb) )
    <=> ( pred_numeral(K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_2017_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa(nat,nat,suc,Nb) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( Nb = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_2018_nth__replicate,axiom,
    ! [A: $tType,I2: nat,Nb: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
     => ( aa(nat,A,nth(A,replicate(A,Nb,Xb)),I2) = Xb ) ) ).

% nth_replicate
tff(fact_2019_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_2020_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_2021_triangle__Suc,axiom,
    ! [Nb: nat] : nat_triangle(aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(Nb)),aa(nat,nat,suc,Nb)) ).

% triangle_Suc
tff(fact_2022_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_2023_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_2024_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(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_2025_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) )
        <=> ( Nb = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2026_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( Nb = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2027_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: 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))),Nb)),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))),Nb)),A2)) = A2 ) ).

% left_minus_one_mult_self
tff(fact_2028_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: 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))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_2029_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_2030_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(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))) = $ite(aa(A,$o,aa(A,fun(A,$o),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,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_2031_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(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)) = $ite(aa(A,$o,aa(A,fun(A,$o),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(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_2032_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(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))) = $ite(aa(A,$o,aa(A,fun(A,$o),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,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_2033_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_2034_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)) ) ).

% diff_numeral_simps(2)
tff(fact_2035_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb))) ) ).

% diff_numeral_simps(3)
tff(fact_2036_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_2037_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_2038_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_2039_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_2040_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_2041_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_2042_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K)),Nb) ) ).

% less_numeral_Suc
tff(fact_2043_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_2044_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit1,K)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_2045_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),Nb) ) ).

% le_numeral_Suc
tff(fact_2046_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_2047_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),Mb) ) ) ).

% neg_numeral_le_iff
tff(fact_2048_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K)),Nb) ).

% diff_numeral_Suc
tff(fact_2049_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_2050_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Nb),Mb) ) ) ).

% neg_numeral_less_iff
tff(fact_2051_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),Nb)) ).

% max_numeral_Suc
tff(fact_2052_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_2053_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),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),Mb)))
        <=> ( Mb != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2054_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),one_one(A)))
        <=> ( Mb != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2055_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_2056_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),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_2057_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 )
        <=> $ite(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))),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2058_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))) )
        <=> $ite(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,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2059_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),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_2060_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),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_2061_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),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_minus
tff(fact_2062_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [P2: $o] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> (P2) ) ) ).

% odd_of_bool_self
tff(fact_2063_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),aa(num,num,bit0,one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_2064_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_2065_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(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),aa(num,num,bit0,one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_2066_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),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% minus_1_div_2_eq
tff(fact_2067_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_2068_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% minus_1_mod_2_eq
tff(fact_2069_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: 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),aa(num,num,bit0,one2))),Nb)) = 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),aa(num,num,bit0,one2))),Nb)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_2070_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_2071_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A2: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% power_minus_odd
tff(fact_2072_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_2073_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),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),Mb),one2))) ) ).

% diff_numeral_special(4)
tff(fact_2074_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb)) ) ).

% diff_numeral_special(3)
tff(fact_2075_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),aa(num,num,bit0,one2))) ) ) ).

% dbl_simps(4)
tff(fact_2076_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: 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),aa(num,num,bit0,one2))),Nb)) = one_one(A) ) ).

% power_minus1_even
tff(fact_2077_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_2078_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_2079_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),aa(num,num,bit0,one2)))) ) ).

% signed_take_bit_0
tff(fact_2080_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_2081_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),aa(num,num,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),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_2082_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_2083_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_2084_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),aa(num,num,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),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_2085_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ).

% one_mod_2_pow_eq
tff(fact_2086_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,aa(int,fun(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),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_2087_signed__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_2088_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_2089_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P)
            & (Q) )) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_conj
tff(fact_2090_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),P2: product_prod(B,C)] :
      ( aa(set(A),$o,member(A,Z),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),P2))
     => ~ ! [X2: B,Y4: C] :
            ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y4) )
           => ~ aa(set(A),$o,member(A,Z),aa(C,set(A),aa(B,fun(C,set(A)),C2,X2),Y4)) ) ) ).

% mem_case_prodE
tff(fact_2091_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_imp_neg_le
tff(fact_2092_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_le_iff
tff(fact_2093_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_minus_iff
tff(fact_2094_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_2095_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_less_iff
tff(fact_2096_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% less_minus_iff
tff(fact_2097_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Mb: num,Nb: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb)) != aa(num,A,numeral_numeral(A),Nb) ) ).

% neg_numeral_neq_numeral
tff(fact_2098_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Mb: num,Nb: num] : aa(num,A,numeral_numeral(A),Mb) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% numeral_neq_neg_numeral
tff(fact_2099_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_2100_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_2101_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_2102_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_2103_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_2104_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_2105_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_diff_commute
tff(fact_2106_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ).

% minus_diff_minus
tff(fact_2107_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_2108_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_2109_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_2110_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_2111_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_2112_mod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A6: A] :
          ( ( modulo_modulo(A,A2,B2) = modulo_modulo(A,A6,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A6),B2) ) ) ) ).

% mod_minus_cong
tff(fact_2113_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_2114_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P2: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C2),P2)
     => ~ ! [X2: A,Y4: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y4) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X2),Y4) ) ) ).

% case_prodE
tff(fact_2115_case__prodD,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2))
     => aa(B,$o,aa(A,fun(B,$o),F2,A2),B2) ) ).

% case_prodD
tff(fact_2116_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,$o))),P2: product_prod(A,B),Z: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C2),P2),Z)
     => ~ ! [X2: A,Y4: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y4) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X2),Y4),Z) ) ) ).

% case_prodE'
tff(fact_2117_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(A,fun(B,fun(C,$o))),A2: A,B2: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R2,A2),B2),C2) ) ).

% case_prodD'
tff(fact_2118_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% zero_less_eq_of_bool
tff(fact_2119_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_2120_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o] :
          aa($o,A,zero_neq_one_of_bool(A),(P2)) = $ite((P2),one_one(A),zero_zero(A)) ) ).

% of_bool_def
tff(fact_2121_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P2: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> ( ( (P2)
             => aa(A,$o,P,one_one(A)) )
            & ( ~ (P2)
             => aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_2122_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P2: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> ~ ( ( (P2)
                & ~ aa(A,$o,P,one_one(A)) )
              | ( ~ (P2)
                & ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_2123_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_le_neg_numeral
tff(fact_2124_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_le_numeral
tff(fact_2125_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_neq_neg_numeral
tff(fact_2126_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_less_numeral
tff(fact_2127_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num,Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2128_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_2129_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_2130_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_2131_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_2132_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_2133_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_2134_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_2135_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_2136_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% less_minus_one_simps(2)
tff(fact_2137_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(4)
tff(fact_2138_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,Xb: 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),Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% numeral_times_minus_swap
tff(fact_2139_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),Nb) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_2140_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% one_neq_neg_numeral
tff(fact_2141_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_2142_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_2143_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Xb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb) = one_one(A) )
        <=> ( ( Xb = one_one(A) )
            | ( Xb = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_2144_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,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_2145_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_2146_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(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_2147_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
         => ( X2 = Xb ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_2148_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
     => ( ! [Y4: A] :
            ( aa(set(A),$o,member(A,Y4),aa(list(A),set(A),set2(A),Xs))
           => ( Y4 = Xb ) )
       => ( Xs = replicate(A,Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_2149_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2150_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2151_real__minus__mult__self__le,axiom,
    ! [U: real,Xb: real] : aa(real,$o,aa(real,fun(real,$o),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),Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_2152_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb) = one_one(int) )
     => ( ( Mb = one_one(int) )
        | ( Mb = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_2153_zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb) = one_one(int) )
    <=> ( ( ( Mb = one_one(int) )
          & ( Nb = one_one(int) ) )
        | ( ( Mb = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( Nb = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_2154_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_2155_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_2156_minus__real__def,axiom,
    ! [Xb: real,Y: real] : aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,uminus_uminus(real),Y)) ).

% minus_real_def
tff(fact_2157_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_zero_le_neg_numeral
tff(fact_2158_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_2159_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_zero_less_neg_numeral
tff(fact_2160_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_2161_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_2162_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_2163_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% less_minus_one_simps(1)
tff(fact_2164_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(3)
tff(fact_2165_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))) ) ).

% not_one_le_neg_numeral
tff(fact_2166_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_2167_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_2168_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Mb)) ) ).

% neg_one_le_numeral
tff(fact_2169_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_2170_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2171_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))) ) ).

% not_one_less_neg_numeral
tff(fact_2172_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_2173_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Mb)) ) ).

% neg_one_less_numeral
tff(fact_2174_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_2175_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)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,uminus_uminus(A),B2),A2 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_2176_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 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),A2 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_2177_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_2178_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_2179_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_2180_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_2181_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_2182_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_2183_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = 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))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_minus
tff(fact_2184_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) ) ).

% power_minus_Bit0
tff(fact_2185_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_2186_real__add__less__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,uminus_uminus(real),Xb)) ) ).

% real_add_less_0_iff
tff(fact_2187_real__0__less__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),Xb)),Y) ) ).

% real_0_less_add_iff
tff(fact_2188_real__add__le__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,uminus_uminus(real),Xb)) ) ).

% real_add_le_0_iff
tff(fact_2189_real__0__le__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xb)),Y) ) ).

% real_0_le_add_iff
tff(fact_2190_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_2191_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),modulo_modulo(int,A2,B2))) ).

% zmod_zminus1_eq_if
tff(fact_2192_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,A2,B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_2193_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),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_2194_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),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_2195_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),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_2196_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),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_2197_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_2198_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_2199_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)) )
        <=> $ite(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),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2200_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) )
        <=> $ite(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,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2201_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: 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) = $ite(Z = zero_zero(A),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_2202_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: 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),Xb),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),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2203_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(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) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(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_2204_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(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_2205_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2206_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ).

% even_minus
tff(fact_2207_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
        <=> ( ( Xb = Y )
            | ( Xb = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_2208_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))) ) ).

% uminus_power_if
tff(fact_2209_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] : aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_2210_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),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_2211_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),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_2212_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),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_2213_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),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_2214_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_2215_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_2216_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),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),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(A,$o),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_2217_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),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)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),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)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),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_2218_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),aa(num,num,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_2219_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% minus_one_power_iff
tff(fact_2220_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( 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),Nb),K)) = 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),minus_minus(nat),Nb),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2221_realpow__square__minus__le,axiom,
    ! [U: real,Xb: real] : aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% realpow_square_minus_le
tff(fact_2222_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2223_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),Nb))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2224_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2225_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),one_one(int))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2226_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2227_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(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_2228_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(modulo_modulo(int,A2,B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(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_2229_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q2: int,R: 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),R))
     => ( ( 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),
                $ite(R = zero_zero(int),aa(int,int,uminus_uminus(int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),
              $ite(R = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R)))) ) ) ).

% zminus1_lemma
tff(fact_2230_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [A4: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A4),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A4 )
             => aa(A,$o,P,A4) )
         => ( ! [A4: A,B3: $o] :
                ( aa(A,$o,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($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A4))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A4 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A4))) ) )
           => aa(A,$o,P,A2) ) ) ) ).

% bits_induct
tff(fact_2231_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),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),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,aa(A,fun(A,$o),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_2232_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),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)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),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)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$o,aa(A,fun(A,$o),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_2233_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_2234_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,Nb: 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)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb)) = 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),aa(num,num,bit0,one2))),Nb)) ) ).

% minus_power_mult_self
tff(fact_2235_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),Nb))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2236_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),Nb))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2237_minus__1__div__exp__eq__int,axiom,
    ! [Nb: 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),aa(num,num,bit0,one2))),Nb)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_2238_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),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_2239_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] : modulo_modulo(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)) ) ).

% exp_mod_exp
tff(fact_2240_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = 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),Mb),Nb)),modulo_modulo(nat,Mb,Nb)) ).

% divmod_nat_def
tff(fact_2241_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: 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),aa(num,num,bit0,one2))),Nb))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_2242_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K2: int] :
              ( aa(int,$o,P,K2)
             => ( ( K2 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) ) )
         => ( ! [K2: int] :
                ( aa(int,$o,P,K2)
               => ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_2243_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),Nb)))
     => aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),aa(nat,nat,suc,Nb)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2244_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb) != zero_zero(A) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))) ) ).

% exp_div_exp_eq
tff(fact_2245_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
        $let(
          half: nat,
          half:= 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),aa(num,num,bit0,one2))),
          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),aa(num,num,bit0,one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= 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),aa(num,num,bit0,one2))),
          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),aa(num,num,bit0,one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_2246_one__div__minus__numeral,axiom,
    ! [Nb: 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),Nb))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Nb))) ).

% one_div_minus_numeral
tff(fact_2247_minus__one__div__numeral,axiom,
    ! [Nb: 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),Nb)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Nb))) ).

% minus_one_div_numeral
tff(fact_2248_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% compl_less_compl_iff
tff(fact_2249_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% compl_le_compl_iff
tff(fact_2250_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_2251_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa($o,int,zero_neq_one_of_bool(int),R != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_2252_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ring_1_of_int(A,K) = $ite(
            K = zero_zero(int),
            zero_zero(A),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
              aa(A,A,uminus_uminus(A),ring_1_of_int(A,aa(int,int,uminus_uminus(int),K))),
              $let(
                l: A,
                l:= aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),
                $ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = zero_zero(int),l,aa(A,A,aa(A,fun(A,A),plus_plus(A),l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_2253_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X4: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bm($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X4)
    <=> ( (P)
        & aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q),X4) ) ) ).

% split_part
tff(fact_2254_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_2255_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,Nb: num] :
          ( ( ring_1_of_int(A,Z) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( Z = aa(num,int,numeral_numeral(int),Nb) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_2256_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : ring_1_of_int(A,aa(num,int,numeral_numeral(int),K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_numeral
tff(fact_2257_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,W)),ring_1_of_int(A,Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% of_int_le_iff
tff(fact_2258_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,W)),ring_1_of_int(A,Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% of_int_less_iff
tff(fact_2259_of__int__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_of_int(A,one_one(int)) = one_one(A) ) ) ).

% of_int_1
tff(fact_2260_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( ring_1_of_int(A,Z) = one_one(A) )
        <=> ( Z = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_2261_of__int__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : 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),ring_1_of_int(A,W)),ring_1_of_int(A,Z)) ) ).

% of_int_mult
tff(fact_2262_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : 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),ring_1_of_int(A,W)),ring_1_of_int(A,Z)) ) ).

% of_int_add
tff(fact_2263_of__int__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),minus_minus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,W)),ring_1_of_int(A,Z)) ) ).

% of_int_diff
tff(fact_2264_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,Nb: nat] : ring_1_of_int(A,aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),ring_1_of_int(A,Z)),Nb) ) ).

% of_int_power
tff(fact_2265_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W: nat,Xb: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),ring_1_of_int(A,B2)),W) = ring_1_of_int(A,Xb) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) = Xb ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_2266_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: int,B2: int,W: nat] :
          ( ( ring_1_of_int(A,Xb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),ring_1_of_int(A,B2)),W) )
        <=> ( Xb = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_2267_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_2268_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_2269_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_2270_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),ring_1_of_int(A,Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_2271_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_2272_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),ring_1_of_int(A,Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_2273_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_2274_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_le_numeral_iff
tff(fact_2275_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),ring_1_of_int(A,Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_2276_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_less_numeral_iff
tff(fact_2277_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),ring_1_of_int(A,Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_2278_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),ring_1_of_int(A,Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_2279_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_2280_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),ring_1_of_int(A,Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_2281_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_2282_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb) = ring_1_of_int(A,Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_2283_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: nat] :
          ( ( ring_1_of_int(A,Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_2284_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),ring_1_of_int(A,B2)),W)),ring_1_of_int(A,Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xb) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2285_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B2: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),ring_1_of_int(A,B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2286_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),ring_1_of_int(A,B2)),W)),ring_1_of_int(A,Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xb) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2287_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B2: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),ring_1_of_int(A,B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2288_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),ring_1_of_int(A,A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2289_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2290_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),ring_1_of_int(A,A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2291_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2292_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: nat] :
          ( ( 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),Xb))),Nb) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_2293_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: 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),Xb))),Nb) = 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),Xb))),Nb) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_2294_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),ring_1_of_int(A,A2))
        <=> aa(int,$o,aa(int,fun(int,$o),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),Xb))),Nb)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2295_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),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),Xb))),Nb)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2296_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),ring_1_of_int(A,A2))
        <=> aa(int,$o,aa(int,fun(int,$o),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),Xb))),Nb)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2297_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),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),Xb))),Nb)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2298_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_bn(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_2299_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),ring_1_of_int(A,Xb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),ring_1_of_int(A,Xb)) ) ).

% mult_of_int_commute
tff(fact_2300_real__of__int__div4,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xb))),aa(real,real,aa(real,fun(real,real),divide_divide(real),ring_1_of_int(real,Nb)),ring_1_of_int(real,Xb))) ).

% real_of_int_div4
tff(fact_2301_real__of__int__div,axiom,
    ! [D2: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),Nb)
     => ( ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),D2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),ring_1_of_int(real,Nb)),ring_1_of_int(real,D2)) ) ) ).

% real_of_int_div
tff(fact_2302_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),ring_1_of_int(A,Z)) ) ) ).

% of_int_nonneg
tff(fact_2303_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),ring_1_of_int(A,Z)) ) ) ).

% of_int_pos
tff(fact_2304_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : 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_2305_int__le__real__less,axiom,
    ! [Nb: int,Mb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),Mb)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),ring_1_of_int(real,Nb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,Mb)),one_one(real))) ) ).

% int_le_real_less
tff(fact_2306_int__less__real__le,axiom,
    ! [Nb: int,Mb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),Mb)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,Nb)),one_one(real))),ring_1_of_int(real,Mb)) ) ).

% int_less_real_le
tff(fact_2307_real__of__int__div__aux,axiom,
    ! [Xb: int,D2: int] : aa(real,real,aa(real,fun(real,real),divide_divide(real),ring_1_of_int(real,Xb)),ring_1_of_int(real,D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),D2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),ring_1_of_int(real,modulo_modulo(int,Xb,D2))),ring_1_of_int(real,D2))) ).

% real_of_int_div_aux
tff(fact_2308_real__of__int__div2,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),ring_1_of_int(real,Nb)),ring_1_of_int(real,Xb))),ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xb)))) ).

% real_of_int_div2
tff(fact_2309_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),ring_1_of_int(real,Nb)),ring_1_of_int(real,Xb))),ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xb)))),one_one(real)) ).

% real_of_int_div3
tff(fact_2310_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),aTP_Lamp_bo(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_2311_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),ring_1_of_int(A,K))
        <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K) ) ) ).

% even_of_int_iff
tff(fact_2312_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : neg_numeral_dbl_dec(A,Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_2313_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% compl_mono
tff(fact_2314_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_2315_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% compl_le_swap2
tff(fact_2316_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Y)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% compl_less_swap2
tff(fact_2317_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_2318_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% diff_shunt_var
tff(fact_2319_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z4: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,Z4)),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z4),one_one(int)))) ) ) ).

% floor_exists
tff(fact_2320_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [X2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,X2)),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int))))
          & ! [Y5: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,Y5)),Xb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),Y5),one_one(int)))) )
             => ( Y5 = X2 ) ) ) ) ).

% floor_exists1
tff(fact_2321_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,uminus_uminus(real),Xb)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2322_divmod__BitM__2__eq,axiom,
    ! [Mb: num] : unique8689654367752047608divmod(int,bitM(Mb),aa(num,num,bit0,one2)) = 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),minus_minus(int),aa(num,int,numeral_numeral(int),Mb)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_2323_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va3: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va3)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va3))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_2324_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_bp(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_2325_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2326_ln__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_2327_ln__inj__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( ( aa(real,real,ln_ln(real),Xb) = aa(real,real,ln_ln(real),Y) )
        <=> ( Xb = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_2328_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_2329_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_2330_ln__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( ( aa(real,real,ln_ln(real),Xb) = zero_zero(real) )
      <=> ( Xb = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_2331_ln__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ).

% ln_gt_zero_iff
tff(fact_2332_ln__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_2333_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit0,K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_2334_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_2335_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_2336_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_2337_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_2338_ln__less__self,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),Xb) ) ).

% ln_less_self
tff(fact_2339_semiring__norm_I28_J,axiom,
    ! [Nb: num] : bitM(aa(num,num,bit1,Nb)) = aa(num,num,bit1,aa(num,num,bit0,Nb)) ).

% semiring_norm(28)
tff(fact_2340_semiring__norm_I27_J,axiom,
    ! [Nb: num] : bitM(aa(num,num,bit0,Nb)) = aa(num,num,bit1,bitM(Nb)) ).

% semiring_norm(27)
tff(fact_2341_ln__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),Xb) ) ).

% ln_bound
tff(fact_2342_ln__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb)) ) ).

% ln_gt_zero
tff(fact_2343_ln__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_2344_ln__gt__zero__imp__gt__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_2345_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb)) ) ).

% ln_ge_zero
tff(fact_2346_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Nb)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(Nb))) ).

% eval_nat_numeral(2)
tff(fact_2347_BitM__plus__one,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(Nb)),one2) = aa(num,num,bit0,Nb) ).

% BitM_plus_one
tff(fact_2348_one__plus__BitM,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(Nb)) = aa(num,num,bit0,Nb) ).

% one_plus_BitM
tff(fact_2349_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_2350_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_2351_ln__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_2352_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( ( aa(real,real,ln_ln(real),Xb) = aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),one_one(real)) )
       => ( Xb = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_2353_ln__div,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_2354_ln__2__less__1,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),one_one(real)) ).

% ln_2_less_1
tff(fact_2355_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),bitM(Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb))),one_one(A)) ) ).

% numeral_BitM
tff(fact_2356_odd__numeral__BitM,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [W: num] : ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,A,numeral_numeral(A),bitM(W))) ) ).

% odd_numeral_BitM
tff(fact_2357_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_2358_ln__diff__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_2359_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_2360_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),Xb))),aa(real,real,uminus_uminus(real),Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_2361_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_12: nat] : aa(nat,$o,P,X_12)
       => ? [N: nat] :
            ( ~ aa(nat,$o,P,N)
            & aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_2362_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z4: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),ring_1_of_int(A,Z4)) ) ).

% ex_le_of_int
tff(fact_2363_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z4: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),ring_1_of_int(A,Z4)) ) ).

% ex_less_of_int
tff(fact_2364_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z4: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,Z4)),Xb) ) ).

% ex_of_int_less
tff(fact_2365_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_2366_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A] : aa(A,A,artanh(A),Xb) = 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)),Xb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% artanh_def
tff(fact_2367_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_bq(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_2368_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(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)),Xb))),Xb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2369_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),ring_1_of_int(A,Y))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
           => ( archimedean_round(A,Xb) = Y ) ) ) ) ).

% round_unique
tff(fact_2370_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2371_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(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)),Xb))),Xb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2372_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Nb) ) ).

% abs_numeral
tff(fact_2373_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_2374_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_2375_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_2376_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_2377_tanh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xb)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% tanh_real_less_iff
tff(fact_2378_tanh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tanh(real),Xb)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% tanh_real_le_iff
tff(fact_2379_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_2380_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% abs_le_self_iff
tff(fact_2381_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2382_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A2))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2383_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),Nb) ) ).

% abs_neg_numeral
tff(fact_2384_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_2385_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: 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)),Nb)) = aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% abs_power_minus
tff(fact_2386_tanh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tanh(real),Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% tanh_real_pos_iff
tff(fact_2387_tanh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_2388_tanh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tanh(real),Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_2389_tanh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tanh(real),Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% tanh_real_nonneg_iff
tff(fact_2390_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),Nb)) = aa(num,int,numeral_numeral(int),Nb) ) ).

% round_numeral
tff(fact_2391_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_2392_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2393_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2394_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2395_artanh__minus__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),Xb)) ) ) ).

% artanh_minus_real
tff(fact_2396_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Nb))
        <=> ( ( A2 != zero_zero(A) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2397_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),aa(num,num,bit0,one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_abs
tff(fact_2398_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),aa(num,num,bit0,one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% abs_power2
tff(fact_2399_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)) ) ).

% round_neg_numeral
tff(fact_2400_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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_2401_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% abs_le_D1
tff(fact_2402_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_self
tff(fact_2403_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_2404_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2405_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ).

% abs_minus_commute
tff(fact_2406_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Nb) ) ).

% power_abs
tff(fact_2407_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,Mb: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),ring_1_of_int(A,archimedean_round(A,Z))))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),ring_1_of_int(A,Mb)))) ) ).

% round_diff_minimal
tff(fact_2408_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_zero
tff(fact_2409_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_2410_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_2411_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),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_2412_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D2)
           => aa(A,$o,aa(A,fun(A,$o),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),D2)) ) ) ) ).

% abs_mult_less
tff(fact_2413_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),B2),A2))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2414_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% abs_triangle_ineq3
tff(fact_2415_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% abs_triangle_ineq2
tff(fact_2416_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_2417_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_minus_self
tff(fact_2418_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_le_iff
tff(fact_2419_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ).

% abs_le_D2
tff(fact_2420_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2) ) ) ) ).

% abs_leI
tff(fact_2421_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_less_iff
tff(fact_2422_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xb)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_2423_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xb: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),E2) )
         => ( Xb = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2424_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),Xb) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb)) ) ) ) ).

% abs_mult_pos
tff(fact_2425_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
              | aa(A,$o,aa(A,fun(A,$o),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_2426_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),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_2427_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2428_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2429_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),Xb)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% abs_div_pos
tff(fact_2430_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),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)),Nb)) ) ).

% zero_le_power_abs
tff(fact_2431_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2432_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),A2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)),aa(A,A,uminus_uminus(A),A2),A2) ) ).

% abs_if
tff(fact_2433_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X4: A] :
          aa(A,A,abs_abs(A),X4) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),zero_zero(A)),aa(A,A,uminus_uminus(A),X4),X4) ) ).

% abs_if_raw
tff(fact_2434_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2))),R)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R)) ) ) ) ).

% abs_diff_le_iff
tff(fact_2435_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(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_2436_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(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),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_2437_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2))),R)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R)) ) ) ) ).

% abs_diff_less_iff
tff(fact_2438_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xb)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_2439_abs__real__def,axiom,
    ! [A2: real] :
      aa(real,real,abs_abs(real),A2) = $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real)),aa(real,real,uminus_uminus(real),A2),A2) ).

% abs_real_def
tff(fact_2440_lemma__interval__lt,axiom,
    ! [A2: real,Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [Y5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y5))),D3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Y5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2441_sin__bound__lemma,axiom,
    ! [Xb: real,Y: real,U: real,V: real] :
      ( ( Xb = Y )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),U)),V)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),U)),Y))),V) ) ) ).

% sin_bound_lemma
tff(fact_2442_tanh__real__gt__neg1,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),Xb)) ).

% tanh_real_gt_neg1
tff(fact_2443_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),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),Xb))) ) ).

% abs_add_one_gt_zero
tff(fact_2444_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),ring_1_of_int(A,Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ) ).

% of_int_leD
tff(fact_2445_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),ring_1_of_int(A,Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ) ).

% of_int_lessD
tff(fact_2446_lemma__interval,axiom,
    ! [A2: real,Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [Y5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y5))),D3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Y5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_2447_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,archimedean_round(A,Xb))),Xb))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% of_int_round_abs_le
tff(fact_2448_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),ring_1_of_int(A,Nb)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))
         => ( archimedean_round(A,Xb) = Nb ) ) ) ).

% round_unique'
tff(fact_2449_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),aa(A,A,abs_abs(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% abs_le_square_iff
tff(fact_2450_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),Xb) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2451_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) ) ) ) ).

% power_even_abs
tff(fact_2452_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),Xb: A] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X2)
             => aa(A,$o,aa(A,fun(A,$o),P,X2),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P,aa(A,A,abs_abs(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_2453_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2454_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_2455_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xb)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2456_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono_even
tff(fact_2457_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(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)),Xb))),Xb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2458_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,archimedean_round(A,Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% of_int_round_le
tff(fact_2459_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),ring_1_of_int(A,archimedean_round(A,Xb))) ) ).

% of_int_round_ge
tff(fact_2460_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),ring_1_of_int(A,archimedean_round(A,Xb))) ) ).

% of_int_round_gt
tff(fact_2461_arctan__double,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,arctan,Xb)) = 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),aa(num,num,bit0,one2))),Xb)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% arctan_double
tff(fact_2462_set__decode__0,axiom,
    ! [Xb: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),nat_set_decode(Xb))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Xb) ) ).

% set_decode_0
tff(fact_2463_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,Nb)),nat_set_decode(Xb))
    <=> aa(set(nat),$o,member(nat,Nb),nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% set_decode_Suc
tff(fact_2464_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R: A,A2: A,B2: A,C2: A,D2: A] :
          ( ( R != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2465_Sum__Icc__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Mb),Nb)
     => ( aa(set(int),int,groups7311177749621191930dd_sum(int,int,aTP_Lamp_br(int,int)),set_or1337092689740270186AtMost(int,Mb,Nb)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),Mb),aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),one_one(int))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ) ) ).

% Sum_Icc_int
tff(fact_2466_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bs(nat,fun(nat,$o),Xb)) ).

% set_decode_def
tff(fact_2467_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_bt(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2468_abs__sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_bu(fun(B,A),fun(B,A),F2)),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_bu(fun(B,A),fun(B,A),F2)),A3) ) ).

% abs_sum_abs
tff(fact_2469_arctan__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_2470_zero__less__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arctan,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% zero_less_arctan_iff
tff(fact_2471_arctan__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_2472_zero__le__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arctan,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% zero_le_arctan_iff
tff(fact_2473_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bv(A,fun(fun(A,B),fun(A,B)),A2),B2)),S) = $ite(aa(set(A),$o,member(A,A2),S),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_2474_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bw(A,fun(fun(A,B),fun(A,B)),A2),B2)),S) = $ite(aa(set(A),$o,member(A,A2),S),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_2475_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_bu(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs
tff(fact_2476_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : 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_bx(fun(B,int),fun(B,A),F2)),A3) ) ).

% of_int_sum
tff(fact_2477_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_bu(fun(B,A),fun(B,A),F2)),A3)) ) ).

% sum_abs_ge_zero
tff(fact_2478_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_by(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_ca(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3)),B4) ) ).

% sum.swap
tff(fact_2479_arctan__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Xb)),aa(real,real,arctan,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% arctan_less_iff
tff(fact_2480_arctan__monotone,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Xb)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone
tff(fact_2481_arctan__monotone_H,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,Xb)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone'
tff(fact_2482_arctan__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arctan,Xb)),aa(real,real,arctan,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% arctan_le_iff
tff(fact_2483_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K5: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),K5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),aa(A,B,G,I3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),K5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),K5)) ) ) ).

% sum_mono
tff(fact_2484_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R: A,F2: fun(B,A),A3: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R),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_cb(A,fun(fun(B,A),fun(B,A)),R),F2)),A3) ) ).

% sum_distrib_left
tff(fact_2485_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),R: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),R) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_cc(fun(B,A),fun(A,fun(B,A)),F2),R)),A3) ) ).

% sum_distrib_right
tff(fact_2486_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B4: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),B4)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_ce(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B4)),A3) ) ).

% sum_product
tff(fact_2487_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_cf(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_2488_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_cg(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3) = aa(A,A,aa(A,fun(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_2489_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),A3: set(B),R: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),A3)),R) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ch(fun(B,A),fun(A,fun(B,A)),F2),R)),A3) ) ).

% sum_divide_distrib
tff(fact_2490_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_ci(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_2491_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),B4: set(B),G: fun(A,fun(B,C)),R2: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(B,B4)
           => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_ck(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B4),G),R2)),A3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_cm(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R2)),B4) ) ) ) ) ).

% sum.swap_restrict
tff(fact_2492_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_cn(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_2493_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_2494_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ).

% sum_nonneg
tff(fact_2495_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F2: fun(B,A),I5: set(B),G: fun(B,A),I2: B] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F2),I5) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),I5) )
         => ( ! [I3: B] :
                ( aa(set(B),$o,member(B,I3),I5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,I3)),aa(B,A,G,I3)) )
           => ( aa(set(B),$o,member(B,I2),I5)
             => ( finite_finite2(B,I5)
               => ( aa(B,A,F2,I2) = aa(B,A,G,I2) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_2496_abs__zmult__eq__1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),Mb) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_2497_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_co(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_cp(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_2498_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3) = zero_zero(B) )
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => ( aa(A,B,F2,X) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_2499_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S3: set(A),Ta: set(B),G: fun(B,C),I2: fun(B,A),F2: fun(A,C)] :
          ( finite_finite2(A,S3)
         => ( finite_finite2(B,Ta)
           => ( ! [X2: B] :
                  ( aa(set(B),$o,member(B,X2),Ta)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G,X2)) )
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),S3)
                   => ? [Xa: B] :
                        ( aa(set(B),$o,member(B,Xa),Ta)
                        & ( aa(B,A,I2,Xa) = X2 )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X2)),aa(B,C,G,Xa)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,groups7311177749621191930dd_sum(A,C,F2),S3)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_2500_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
           => ( ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,G,X4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_2501_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R2: fun(A,fun(A,$o)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,zero_zero(A)),zero_zero(A))
         => ( ! [X15: A,Y15: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R2,X15),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R2,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X22),Y23)) )
           => ( finite_finite2(B,S)
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),S)
                   => aa(A,$o,aa(A,fun(A,$o),R2,aa(B,A,H,X2)),aa(B,A,G,X2)) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S)) ) ) ) ) ) ).

% sum.related
tff(fact_2502_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum_strict_mono
tff(fact_2503_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T3: set(B),S: set(A),I2: fun(B,A),J: fun(A,B),T2: set(B),G: fun(A,C),H: fun(B,C)] :
          ( finite_finite2(A,S4)
         => ( finite_finite2(B,T3)
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S4))
                 => ( aa(B,A,I2,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S4))
                   => aa(set(B),$o,member(B,aa(A,B,J,A4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3)) )
               => ( ! [B3: B] :
                      ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                     => ( aa(A,B,J,aa(B,A,I2,B3)) = B3 ) )
                 => ( ! [B3: B] :
                        ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                       => aa(set(A),$o,member(A,aa(B,A,I2,B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S4)) )
                   => ( ! [A4: A] :
                          ( aa(set(A),$o,member(A,A4),S4)
                         => ( aa(A,C,G,A4) = zero_zero(C) ) )
                     => ( ! [B3: B] :
                            ( aa(set(B),$o,member(B,B3),T3)
                           => ( aa(B,C,H,B3) = zero_zero(C) ) )
                       => ( ! [A4: A] :
                              ( aa(set(A),$o,member(A,A4),S)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
                         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,G),S) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,H),T2) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_2504_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S3: set(A),F2: fun(A,B),B4: B,I2: A] :
          ( finite_finite2(A,S3)
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S3) = B4 )
             => ( aa(set(A),$o,member(A,I2),S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),B4) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2505_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S3: set(A),F2: fun(A,B),I2: A] :
          ( finite_finite2(A,S3)
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S3) = zero_zero(B) )
             => ( aa(set(A),$o,member(A,I2),S3)
               => ( aa(A,B,F2,I2) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2506_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,aa(int,fun(int,$o),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_2507_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cq(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2508_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I2: A,F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( aa(set(A),$o,member(A,I2),I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I2))
             => ( ! [I3: A] :
                    ( aa(set(A),$o,member(A,I3),I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_2509_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( aa(set(A),$o,member(A,I3),I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) ) ) ) ) ).

% sum_pos
tff(fact_2510_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C5)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B3: A] :
                      ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B4))
                     => ( aa(A,B,H,B3) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B4) )
                  <=> ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),C5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C5) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_2511_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C5)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B3: A] :
                      ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B4))
                     => ( aa(A,B,H,B3) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),C5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C5) )
                   => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B4) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_2512_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T2) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_2513_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T2) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_2514_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,H,X2) = zero_zero(B) ) )
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),S)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_2515_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),S)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T2) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_2516_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B4: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
         => ( finite_finite2(A,A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B4)) ) ) ) ) ).

% sum.subset_diff
tff(fact_2517_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),B4: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B4)) ) ) ) ) ).

% sum_diff
tff(fact_2518_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(Mb)),nat_set_decode(Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_2519_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb)),Mb)
      <=> ( aa(int,int,abs_abs(int),Nb) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_2520_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B4: set(A),A3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
           => ( ! [B3: A] :
                  ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,B3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B4)) ) ) ) ) ).

% sum_mono2
tff(fact_2521_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,abs_abs(int),L)))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_add_abs_iff
tff(fact_2522_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),K)),L))
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_abs_add_iff
tff(fact_2523_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B4: set(A),A3: set(A),B2: A,F2: fun(A,B)] :
          ( finite_finite2(A,B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
           => ( aa(set(A),$o,member(A,B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,B2))
               => ( ! [X2: A] :
                      ( aa(set(A),$o,member(A,X2),B4)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
                 => aa(B,$o,aa(B,fun(B,$o),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_2524_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xb: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xb,I3)) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xb),I5) = one_one(B) )
           => ( ! [I3: A] :
                  ( aa(set(A),$o,member(A,I3),I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,I3)),B2))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_cr(fun(A,B),fun(fun(A,B),fun(A,B)),Xb),A2)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2525_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),I3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,Mb)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
           => ? [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),I3)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
                & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2526_incr__lemma,axiom,
    ! [D2: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),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,aa(int,fun(int,int),minus_minus(int),Xb),Z))),one_one(int))),D2))) ) ).

% incr_lemma
tff(fact_2527_decr__lemma,axiom,
    ! [D2: int,Xb: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),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,aa(int,fun(int,int),minus_minus(int),Xb),Z))),one_one(int))),D2))),Z) ) ).

% decr_lemma
tff(fact_2528_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_2529_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y: A,Xb: 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),Xb),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),Xb),Y)) )
        <=> ( ( W = Xb )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_2530_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D2 ) )
        <=> ( 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),D2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_2531_nat__ivt__aux,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,suc,I3))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2532_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F2: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F2,I3)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Nb))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
              & ( aa(nat,int,F2,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2533_arctan__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),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,Xb)),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),Xb),Y)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_2534_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( finite_finite2(nat,A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(nat),nat,nat_set_encode,A3))
      <=> ~ aa(set(nat),$o,member(nat,zero_zero(nat)),A3) ) ) ).

% even_set_encode_iff
tff(fact_2535_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),Nb)) = 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),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,pred_numeral(Nb)))) ) ).

% mask_numeral
tff(fact_2536_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_2537_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = $ite(Nb = zero_zero(nat),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),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% take_bit_rec
tff(fact_2538_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : size_num(aa(num,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_2539_tanh__real__altdef,axiom,
    ! [Xb: real] : aa(real,real,tanh(real),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(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),aa(num,num,bit0,one2)))),Xb)))),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),aa(num,num,bit0,one2)))),Xb)))) ).

% tanh_real_altdef
tff(fact_2540_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ).

% mask_nat_positive_iff
tff(fact_2541_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_2542_exp__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xb)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% exp_less_cancel_iff
tff(fact_2543_exp__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xb)),aa(real,real,exp(real),Y)) ) ).

% exp_less_mono
tff(fact_2544_exp__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xb)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% exp_le_cancel_iff
tff(fact_2545_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_concat_bit(Nb,K),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) ).

% concat_bit_of_zero_2
tff(fact_2546_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_2547_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_2548_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_2549_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_2550_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( ( bit_se2239418461657761734s_mask(A,Nb) = zero_zero(A) )
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_2551_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_2552_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,exp(real),Xb) = one_one(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_2553_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),one_one(A)) = zero_zero(A) )
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_2554_mask__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% mask_Suc_0
tff(fact_2555_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,exp(real),Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% one_less_exp_iff
tff(fact_2556_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_2557_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_2558_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,exp(real),Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% one_le_exp_iff
tff(fact_2559_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,Nb) ) ).

% take_bit_minus_one_eq_mask
tff(fact_2560_exp__ln,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,exp(real),aa(real,real,ln_ln(real),Xb)) = Xb ) ) ).

% exp_ln
tff(fact_2561_exp__ln__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,exp(real),aa(real,real,ln_ln(real),Xb)) = Xb )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% exp_ln_iff
tff(fact_2562_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ).

% take_bit_of_Suc_0
tff(fact_2563_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Mb),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ).

% sum.cl_ivl_Suc
tff(fact_2564_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ).

% take_bit_of_1
tff(fact_2565_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cs(fun(nat,A),fun(nat,A),C2)),A3) = $ite(
            ( finite_finite2(nat,A3)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A3) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_2566_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))
        <=> ( ( Nb = zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_2567_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D2: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ct(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = $ite(
            ( finite_finite2(nat,A3)
            & aa(set(nat),$o,member(nat,zero_zero(nat)),A3) ),
            aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D2,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_2568_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),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_0
tff(fact_2569_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: nat,Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) ) ).

% take_bit_of_exp
tff(fact_2570_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_of_2
tff(fact_2571_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),ring_1_of_int(A,K)) = ring_1_of_int(A,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_of_int
tff(fact_2572_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : ring_1_of_int(A,bit_se2239418461657761734s_mask(int,Nb)) = bit_se2239418461657761734s_mask(A,Nb) ) ).

% of_int_mask_eq
tff(fact_2573_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ).

% take_bit_add
tff(fact_2574_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A,Mb: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_2575_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_2576_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Mb),Q2)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_2577_exp__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xb)),aa(real,real,exp(real),Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% exp_less_cancel
tff(fact_2578_take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_2579_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_2580_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% take_bit_mult
tff(fact_2581_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)) ).

% take_bit_diff
tff(fact_2582_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B2: int] : bit_concat_bit(Nb,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),B2)) = bit_concat_bit(Nb,B2) ).

% concat_bit_take_bit_eq
tff(fact_2583_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K: int,L: int,R: int,S3: int] :
      ( ( aa(int,int,bit_concat_bit(Nb,K),L) = aa(int,int,bit_concat_bit(Nb,R),S3) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),R) )
        & ( L = S3 ) ) ) ).

% concat_bit_eq_iff
tff(fact_2584_less__eq__mask,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),bit_se2239418461657761734s_mask(nat,Nb)) ).

% less_eq_mask
tff(fact_2585_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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_2586_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X2: real] : aa(real,real,exp(real),X2) = Y ) ).

% exp_total
tff(fact_2587_exp__gt__zero,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,exp(real),Xb)) ).

% exp_gt_zero
tff(fact_2588_not__exp__less__zero,axiom,
    ! [Xb: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,exp(real),Xb)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_2589_exp__ge__zero,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,exp(real),Xb)) ).

% exp_ge_zero
tff(fact_2590_not__exp__le__zero,axiom,
    ! [Xb: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xb)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_2591_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),A3)
         => ( ! [X2: nat] :
                ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,X2)),A3)
               => ( aa(nat,A,F2,aa(nat,nat,suc,X2)) = aa(nat,A,G,aa(nat,nat,suc,X2)) ) )
           => ( 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_2592_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2593_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),B2) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_2594_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ).

% take_bit_nonnegative
tff(fact_2595_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_2596_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_2597_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_2598_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),bit_se2584673776208193580ke_bit(A,Nb),bit_ri4674362597316999326ke_bit(A,Mb)),
            A2) ) ).

% signed_take_bit_take_bit
tff(fact_2599_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) )
         => ( aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),Y)) ) ) ) ).

% exp_add_commuting
tff(fact_2600_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),Y)) = aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ).

% mult_exp_exp
tff(fact_2601_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),Y)) ) ).

% exp_diff
tff(fact_2602_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Mb),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_unset_bit_eq
tff(fact_2603_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Mb),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_set_bit_eq
tff(fact_2604_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se8732182000553998342ip_bit(A,Mb,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se8732182000553998342ip_bit(A,Mb,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_flip_bit_eq
tff(fact_2605_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),aa(num,num,bit0,one2)))) ).

% set_encode_def
tff(fact_2606_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G,X2)),aa(A,nat,F2,X2)) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_cu(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A3) = aa(nat,nat,aa(nat,fun(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_2607_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cv(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2608_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,Nb)) ).

% mask_nonnegative_int
tff(fact_2609_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2239418461657761734s_mask(int,Nb)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_2610_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2611_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
            & ( aa(A,nat,F2,X) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A3)
               => ( ( X != Xa2 )
                 => ( aa(A,nat,F2,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_2612_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),Nb: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = aa(nat,nat,suc,Nb) )
     => ? [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X2)) ) ) ).

% sum_SucD
tff(fact_2613_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3) = one_one(nat) )
      <=> ? [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
            & ( aa(A,nat,F2,X) = one_one(nat) )
            & ! [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A3)
               => ( ( X != Xa2 )
                 => ( aa(A,nat,F2,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2614_exp__gt__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,exp(real),Xb)) ) ).

% exp_gt_one
tff(fact_2615_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2616_exp__ge__add__one__self,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb)),aa(real,real,exp(real),Xb)) ).

% exp_ge_add_one_self
tff(fact_2617_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_2618_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_2619_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,I5: set(nat)] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cx(A,fun(nat,fun(nat,A)),Xb),Mb)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),I5)) ) ).

% sum_power_add
tff(fact_2620_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Nb,Mb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_cy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2621_less__mask,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),bit_se2239418461657761734s_mask(nat,Nb)) ) ).

% less_mask
tff(fact_2622_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_cz(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_da(nat,fun(complex,$o),Nb))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_2623_sum__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Nb)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_cz(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_aw(nat,fun(complex,fun(complex,$o)),Nb),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_2624_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_2625_sum__diff__nat,axiom,
    ! [A: $tType,B4: set(A),A3: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(nat,nat,aa(nat,fun(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_2626_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_2627_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb)),aa(real,real,exp(real),Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_2628_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2629_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),one_one(real)))
          & ( aa(real,real,exp(real),X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_2630_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2631_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2632_ln__ge__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_2633_ln__x__over__x__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),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),Xb)),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_2634_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cv(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2635_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_db(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,aa(nat,nat,suc,Nb))),aa(nat,A,F2,Mb)) ) ) ) ).

% sum_Suc_diff
tff(fact_2636_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_bit0
tff(fact_2637_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) ) ).

% take_bit_eq_mod
tff(fact_2638_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb) = Mb )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2639_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_2640_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_2641_take__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb) = modulo_modulo(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% take_bit_nat_def
tff(fact_2642_exp__le,axiom,
    aa(real,$o,aa(real,fun(real,$o),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_2643_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2644_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% take_bit_int_less_exp
tff(fact_2645_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = modulo_modulo(int,K,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% take_bit_int_def
tff(fact_2646_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tanh(A),Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb)))) ) ).

% tanh_altdef
tff(fact_2647_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_2648_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = zero_zero(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)),A2) ) ) ).

% take_bit_eq_0_iff
tff(fact_2649_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),aa(num,num,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),aa(num,num,bit0,one2))) ) ).

% take_bit_numeral_bit0
tff(fact_2650_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb)),Mb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_2651_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) ).

% Suc_mask_eq_exp
tff(fact_2652_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2239418461657761734s_mask(nat,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% mask_nat_less_exp
tff(fact_2653_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% exp_half_le2
tff(fact_2654_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_2655_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2656_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_2657_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),aa(num,num,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),aa(num,num,bit0,one2))) ) ).

% exp_double
tff(fact_2658_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dc(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Mb)),aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_2659_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Mb)) ) ) ) ).

% sum_telescope''
tff(fact_2660_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,Nb))
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_2661_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2662_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2663_mask__nat__def,axiom,
    ! [Nb: nat] : bit_se2239418461657761734s_mask(nat,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),one_one(nat)) ).

% mask_nat_def
tff(fact_2664_mask__half__int,axiom,
    ! [Nb: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,Nb)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ).

% mask_half_int
tff(fact_2665_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),aa(num,num,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),aa(num,num,bit0,one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_2666_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),one_one(int)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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,Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_2667_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)),one_one(A)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),Nb))) ) ).

% mask_eq_sum_exp
tff(fact_2668_mask__int__def,axiom,
    ! [Nb: nat] : bit_se2239418461657761734s_mask(int,Nb) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),one_one(int)) ).

% mask_int_def
tff(fact_2669_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Mb: nat,Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Mb,Nb))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb))) ) ) ) ).

% sum_gp_multiplied
tff(fact_2670_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: 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),aa(num,num,bit0,one2))),Mb),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_de(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.in_pairs
tff(fact_2671_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),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,Nb),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_2672_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,suc,Nb))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_2673_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,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_2674_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),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,Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% take_bit_Suc
tff(fact_2675_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se2239418461657761734s_mask(A,Nb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_2676_exp__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xb)),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)),Xb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% exp_bound
tff(fact_2677_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_2678_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),Nb))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_2679_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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),aa(num,num,bit0,one2))),Nb)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_2680_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),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),aa(num,num,bit0,one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_2681_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2682_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_df(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% gauss_sum_nat
tff(fact_2683_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),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_2684_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,exp(real),Xb)),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),aa(num,num,bit0,one2))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_2685_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_2686_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dg(nat,fun(nat,fun(nat,nat)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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,Nb)),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),aa(num,num,bit0,one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),D2)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% arith_series_nat
tff(fact_2687_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_df(nat,nat)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Icc_nat
tff(fact_2688_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),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)),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,exp(real),Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_2689_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),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_2690_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_2691_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),
            zero_zero(A),
            $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))),Mb)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ) ).

% sum_gp
tff(fact_2692_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),Xb) = 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),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq1
tff(fact_2693_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = 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),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_2694_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) ) ),
        zero_zero(int),
        $ite(
          K = aa(int,int,uminus_uminus(int),one_one(int)),
          L,
          $ite(L = aa(int,int,uminus_uminus(int),one_one(int)),K,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),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% and_int_unfold
tff(fact_2695_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Mb: nat,Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Mb) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( Mb = Nb ) ) ) ).

% of_nat_eq_iff
tff(fact_2696_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_2697_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_2698_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_2699_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),Mb) = aa(num,int,numeral_numeral(int),V) )
    <=> ( Mb = aa(num,nat,numeral_numeral(nat),V) ) ) ).

% int_eq_iff_numeral
tff(fact_2700_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_2701_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_2702_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Xb) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_2703_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_2704_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,A,semiring_1_of_nat(A),Nb) ) ).

% abs_of_nat
tff(fact_2705_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),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,Nb),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2)) ) ).

% take_bit_and
tff(fact_2706_height__double__log__univ__size,axiom,
    ! [U: real,Dega: nat,Ta: vEBT_VEBT] :
      ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Dega) )
     => ( vEBT_invar_vebt(Ta,Dega)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_2707_of__nat__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) ) ) ).

% of_nat_0
tff(fact_2708_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( zero_zero(nat) = Nb ) ) ) ).

% of_nat_0_eq_iff
tff(fact_2709_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Mb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Mb) = zero_zero(A) )
        <=> ( Mb = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_2710_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% of_nat_less_iff
tff(fact_2711_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,A,numeral_numeral(A),Nb) ) ).

% of_nat_numeral
tff(fact_2712_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% of_nat_le_iff
tff(fact_2713_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_add
tff(fact_2714_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Nb) = one_one(A) )
        <=> ( Nb = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_2715_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( Nb = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_2716_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_2717_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_mult
tff(fact_2718_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_2719_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_2720_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = Xb ) ).

% bit.conj_one_right
tff(fact_2721_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xb: nat,B2: nat,W: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Xb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) )
        <=> ( Xb = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_2722_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B2: nat,W: nat,Xb: 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),Xb) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) = Xb ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_2723_of__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Mb),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),Mb)),Nb) ) ).

% of_nat_power
tff(fact_2724_negative__zless,axiom,
    ! [Nb: nat,Mb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)))),aa(nat,int,semiring_1_of_nat(int),Mb)) ).

% negative_zless
tff(fact_2725_log__one,axiom,
    ! [A2: real] : aa(real,real,log(A2),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_2726_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_2727_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_2728_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_2729_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o] : aa(nat,A,semiring_1_of_nat(A),aa($o,nat,zero_neq_one_of_bool(nat),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% of_nat_of_bool
tff(fact_2730_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_dh(fun(B,nat),fun(B,A),F2)),A3) ) ).

% of_nat_sum
tff(fact_2731_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Mb)),zero_zero(A))
        <=> ( Mb = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_2732_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Mb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Mb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Mb)) ) ).

% of_nat_Suc
tff(fact_2733_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_2734_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_2735_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(num,real,numeral_numeral(real),W))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(num,nat,numeral_numeral(nat),W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_2736_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(num,real,numeral_numeral(real),W)),aa(nat,real,semiring_1_of_nat(real),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_2737_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Mb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_2738_log__eq__one,axiom,
    ! [A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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_2739_log__less__cancel__iff,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_2740_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xb)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_2741_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,log(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_2742_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xb)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_2743_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,log(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_2744_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ).

% of_nat_0_less_iff
tff(fact_2745_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_2746_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),aa(num,num,bit0,Y))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_2747_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_2748_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B2: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_2749_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xb) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_2750_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xb: num,Nb: nat,Y: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb) = 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),Xb)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_2751_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,Xb: num,Nb: 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),Xb)),Nb) )
        <=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_2752_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B2: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2753_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xb) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2754_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,log(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_2755_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xb)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_2756_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,log(A2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_2757_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xb)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_2758_log__le__cancel__iff,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_2759_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Mb))) ) ).

% add_neg_numeral_special(6)
tff(fact_2760_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: 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),Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) ) ).

% add_neg_numeral_special(5)
tff(fact_2761_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) ) ).

% diff_numeral_special(5)
tff(fact_2762_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(Mb)) ) ).

% diff_numeral_special(6)
tff(fact_2763_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),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_2764_and__minus__numerals_I6_J,axiom,
    ! [Nb: 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,Nb)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_2765_and__minus__numerals_I2_J,axiom,
    ! [Nb: 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,Nb)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_2766_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),Xb)),Nb))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xb)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2767_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_2768_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_2769_and__minus__numerals_I5_J,axiom,
    ! [Nb: 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,bit0,Nb)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_2770_and__minus__numerals_I1_J,axiom,
    ! [Nb: 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,bit0,Nb)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_2771_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) ) ) ).

% even_of_nat
tff(fact_2772_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I2: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),Nb)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2773_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),Nb)),Xb) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2774_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I2: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),Nb)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2775_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I2)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I2)),Nb)),Xb) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2776_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_2777_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_2778_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_2779_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_and_eq
tff(fact_2780_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_2781_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : 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),ring_1_of_int(A,K)),ring_1_of_int(A,L)) ) ).

% of_int_and_eq
tff(fact_2782_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% real_arch_simple
tff(fact_2783_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_2784_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),Xb)) ) ).

% mult_of_nat_commute
tff(fact_2785_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(nat,A,semiring_1_of_nat(A),Mb)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Mb)) ) ).

% take_bit_of_nat
tff(fact_2786_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M: nat,N: nat] : Z != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_diff_cases
tff(fact_2787_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se2239418461657761734s_mask(nat,Nb)) = bit_se2239418461657761734s_mask(A,Nb) ) ).

% of_nat_mask_eq
tff(fact_2788_less__log__of__power,axiom,
    ! [B2: real,Nb: nat,Mb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb)),Mb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Mb)) ) ) ).

% less_log_of_power
tff(fact_2789_log__of__power__eq,axiom,
    ! [Mb: nat,B2: real,Nb: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),Mb) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,real,semiring_1_of_nat(real),Nb) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),Mb)) ) ) ) ).

% log_of_power_eq
tff(fact_2790_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Nb)),ring_1_of_int(A,Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Nb)),Xb) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2791_num__induct,axiom,
    ! [P: fun(num,$o),Xb: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X2: num] :
            ( aa(num,$o,P,X2)
           => aa(num,$o,P,inc(X2)) )
       => aa(num,$o,P,Xb) ) ) ).

% num_induct
tff(fact_2792_add__inc,axiom,
    ! [Xb: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y)) ).

% add_inc
tff(fact_2793_le__log__of__power,axiom,
    ! [B2: real,Nb: nat,Mb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb)),Mb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Mb)) ) ) ).

% le_log_of_power
tff(fact_2794_log__base__pow,axiom,
    ! [A2: real,Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( aa(real,real,log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),Nb)),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),Xb)),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log_base_pow
tff(fact_2795_log__nat__power,axiom,
    ! [Xb: real,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,log(B2),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Xb)) ) ) ).

% log_nat_power
tff(fact_2796_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_2797_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_0_le_iff
tff(fact_2798_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Mb)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_2799_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Nb)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_2800_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,Mb: nat,Nb: 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),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))) = 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),Mb))),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% div_mult2_eq'
tff(fact_2801_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% of_nat_less_imp_less
tff(fact_2802_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% less_imp_of_nat_less
tff(fact_2803_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I2: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I2)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_2804_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_2805_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_2806_AND__upper2,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_2807_AND__upper1,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Xb) ) ).

% AND_upper1
tff(fact_2808_AND__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)) ) ).

% AND_lower
tff(fact_2809_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_2810_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),Nb) ) ) ).

% of_nat_dvd_iff
tff(fact_2811_take__bit__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se2239418461657761734s_mask(A,Nb)) ) ).

% take_bit_eq_mask
tff(fact_2812_int__ops_I3_J,axiom,
    ! [Nb: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,int,numeral_numeral(int),Nb) ).

% int_ops(3)
tff(fact_2813_int__of__nat__induct,axiom,
    ! [P: fun(int,$o),Z: int] :
      ( ! [N: nat] : aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),N))
     => ( ! [N: nat] : aa(int,$o,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))
       => aa(int,$o,P,Z) ) ) ).

% int_of_nat_induct
tff(fact_2814_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% int_cases
tff(fact_2815_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),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_2816_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),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_2817_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% zle_int
tff(fact_2818_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_2819_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,Mb,Nb)) = modulo_modulo(A,aa(nat,A,semiring_1_of_nat(A),Mb),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_mod
tff(fact_2820_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_2821_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(nat,int,semiring_1_of_nat(int),Mb)) ).

% int_plus
tff(fact_2822_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),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),Mb),Nb))),Z) ).

% zadd_int_left
tff(fact_2823_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_2824_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z)
    <=> ? [N4: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N4)) ) ).

% zle_iff_zadd
tff(fact_2825_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_2826_int__sum,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)) = aa(set(A),int,groups7311177749621191930dd_sum(A,int,aTP_Lamp_di(fun(A,nat),fun(A,int),F2)),A3) ).

% int_sum
tff(fact_2827_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_2828_log2__of__power__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) )
     => ( aa(nat,real,semiring_1_of_nat(real),Nb) = aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Mb)) ) ) ).

% log2_of_power_eq
tff(fact_2829_log__of__power__less,axiom,
    ! [Mb: nat,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Mb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),Mb))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_less
tff(fact_2830_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),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_2831_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),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_2832_inc_Osimps_I1_J,axiom,
    inc(one2) = aa(num,num,bit0,one2) ).

% inc.simps(1)
tff(fact_2833_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : inc(aa(num,num,bit1,Xb)) = aa(num,num,bit0,inc(Xb)) ).

% inc.simps(3)
tff(fact_2834_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : inc(aa(num,num,bit0,Xb)) = aa(num,num,bit1,Xb) ).

% inc.simps(2)
tff(fact_2835_log__of__power__le,axiom,
    ! [Mb: nat,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Mb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),Mb))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_le
tff(fact_2836_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),Xb)) ) ) ).

% ex_less_of_nat_mult
tff(fact_2837_add__One,axiom,
    ! [Xb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2) = inc(Xb) ).

% add_One
tff(fact_2838_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_2839_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_2840_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_2841_inc__BitM__eq,axiom,
    ! [Nb: num] : inc(bitM(Nb)) = aa(num,num,bit0,Nb) ).

% inc_BitM_eq
tff(fact_2842_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Nb: nat,Mb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ) ).

% of_nat_diff
tff(fact_2843_BitM__inc__eq,axiom,
    ! [Nb: num] : bitM(inc(Nb)) = aa(num,num,bit1,Nb) ).

% BitM_inc_eq
tff(fact_2844_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xb: 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),Nb)),Xb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xb)),Nb) ) ).

% exp_of_nat_mult
tff(fact_2845_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xb)),Nb) ) ).

% exp_of_nat2_mult
tff(fact_2846_log__ln,axiom,
    ! [Xb: real] : aa(real,real,ln_ln(real),Xb) = aa(real,real,log(aa(real,real,exp(real),one_one(real))),Xb) ).

% log_ln
tff(fact_2847_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ! [Y5: real] :
        ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xb)) ) ).

% reals_Archimedean3
tff(fact_2848_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N: nat] : Mb != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( Mb != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ) ) ).

% int_cases4
tff(fact_2849_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xb))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xb))) ).

% real_of_nat_div4
tff(fact_2850_int__Suc,axiom,
    ! [Nb: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) ).

% int_Suc
tff(fact_2851_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_2852_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z)
    <=> ? [N4: 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,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_2853_real__of__nat__div,axiom,
    ! [D2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D2),Nb)
     => ( aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),D2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),D2)) ) ) ).

% real_of_nat_div
tff(fact_2854_mult__inc,axiom,
    ! [Xb: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),Xb),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),Xb),Y)),Xb) ).

% mult_inc
tff(fact_2855_less__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),Mb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Mb))) ) ).

% less_log2_of_power
tff(fact_2856_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),Mb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Mb))) ) ).

% le_log2_of_power
tff(fact_2857_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2) ) ) ) ).

% even_and_iff
tff(fact_2858_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Mb))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_less
tff(fact_2859_log__base__change,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(B2),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_2860_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
        | aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ) ) ).

% even_and_iff_int
tff(fact_2861_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xb: num] : aa(num,A,numeral_numeral(A),inc(Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) ) ).

% numeral_inc
tff(fact_2862_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,Mb: nat,Nb: nat] : modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))) = 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),Mb)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)))),modulo_modulo(A,A2,aa(nat,A,semiring_1_of_nat(A),Mb))) ) ).

% mod_mult2_eq'
tff(fact_2863_succ__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c2(Ta,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_2864_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,Mb,Nb)))),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% field_char_0_class.of_nat_div
tff(fact_2865_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2866_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ~ ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% pos_int_cases
tff(fact_2867_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) )
       => ~ ! [N: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% int_cases3
tff(fact_2868_nat__less__real__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Nb)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),Mb)) ) ).

% nat_less_real_le
tff(fact_2869_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Mb)),one_one(real))) ) ).

% nat_le_real_less
tff(fact_2870_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I2)),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_2871_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),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,Nb)))) ).

% not_zle_0_negative
tff(fact_2872_negative__zless__0,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_2873_negD,axiom,
    ! [Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),zero_zero(int))
     => ? [N: nat] : Xb = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% negD
tff(fact_2874_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Mb))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log2_of_power_le
tff(fact_2875_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)),zero_zero(int),aa(int,int,aa(int,fun(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_2876_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D2: nat] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Xb)),aa(nat,real,semiring_1_of_nat(real),D2)) = 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),Xb),D2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),modulo_modulo(nat,Xb,D2))),aa(nat,real,semiring_1_of_nat(real),D2))) ).

% real_of_nat_div_aux
tff(fact_2877_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),aa(num,num,bit0,one2))) ) ).

% and_one_eq
tff(fact_2878_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),aa(num,num,bit0,one2))) ) ).

% one_and_eq
tff(fact_2879_log__mult,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_2880_log__divide,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(A2),Xb)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_2881_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
         => ~ ! [N: nat] : ~ aa(A,$o,aa(A,fun(A,$o),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,N)))),E) ) ) ).

% nat_approx_posE
tff(fact_2882_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) ) ).

% of_nat_less_two_power
tff(fact_2883_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: nat,Mb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => ( ( Nb != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(A,$o),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),Mb))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ) ) ).

% inverse_of_nat_le
tff(fact_2884_insert__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(Ta,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,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,aa(num,num,bit0,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_2885_succ__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(Ta,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,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,aa(num,num,bit0,aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_2886_member__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(Ta,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),aa(num,num,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),aa(num,num,bit0,one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_2887_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( 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),Xb),aa(nat,A,semiring_1_of_nat(A),Nb)))),Nb) = aa(A,A,exp(A),Xb) ) ) ) ).

% exp_divide_power_eq
tff(fact_2888_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
       => ( ! [M: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),Xb)),C2) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2889_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ~ ! [N: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% neg_int_cases
tff(fact_2890_zdiff__int__split,axiom,
    ! [P: fun(int,$o),Xb: nat,Y: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Y)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xb)
         => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Xb)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_2891_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xb))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xb)))) ).

% real_of_nat_div2
tff(fact_2892_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xb))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xb)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_2893_ln__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_realpow
tff(fact_2894_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
         => ( ( B2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
             => ( aa(real,real,log(A2),Xb) = 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),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2895_linear__plus__1__le__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),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),Nb)),Xb)),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),Xb),one_one(real))),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_2896_Bernoulli__inequality,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => aa(real,$o,aa(real,fun(real,$o),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),Nb)),Xb))),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)),Xb)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_2897_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($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
            & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% and_int_rec
tff(fact_2898_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D2: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_dj(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) = 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),Nb)),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),aa(num,num,bit0,one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),D2))) ) ).

% double_arith_series
tff(fact_2899_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_2900_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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)),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_2901_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum
tff(fact_2902_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D2: A,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_dk(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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),Nb)),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),aa(num,num,bit0,one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),D2)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% arith_series
tff(fact_2903_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) = $ite(Xb = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)),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),Xb),Mb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ).

% sum_gp_offset
tff(fact_2904_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => aa(real,$o,aa(real,fun(real,$o),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),Nb)),Xb))),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)),Xb)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_2905_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Nb))),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),aa(real,real,exp(real),Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_2906_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),aa(real,real,exp(real),aa(real,real,uminus_uminus(real),Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_2907_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq2
tff(fact_2908_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          aa(nat,A,semiring_1_of_nat(A),Nb) = $ite(Nb = zero_zero(nat),zero_zero(A),aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_dl(nat,fun(nat,A))),divmod_nat(Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% of_nat_code_if
tff(fact_2909_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_dm(real,fun(nat,real),Xb)) ) ).

% monoseq_arctan_series
tff(fact_2910_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K5: real,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K5)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(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)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),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),Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2911_log__ceil__idem,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Xb)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),ring_1_of_int(real,archimedean_ceiling(real,Xb)))) ) ) ).

% log_ceil_idem
tff(fact_2912_ln__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
       => ( aa(real,real,ln_ln(real),Xb) = suminf(real,aTP_Lamp_dn(real,fun(nat,real),Xb)) ) ) ) ).

% ln_series
tff(fact_2913_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(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)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),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_dp(A,fun(A,fun(nat,fun(nat,A))),H),Z),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_2914_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( aa(set(A),$o,member(A,I2),aa(A,set(A),set_ord_lessThan(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),K) ) ) ).

% lessThan_iff
tff(fact_2915_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( ring_1_of_int(A,archimedean_ceiling(A,Xb)) = Xb )
        <=> ? [N4: int] : Xb = ring_1_of_int(A,N4) ) ) ).

% of_int_ceiling_cancel
tff(fact_2916_heigt__uplog__rel,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ).

% heigt_uplog_rel
tff(fact_2917_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),Xb)),aa(A,set(A),set_ord_lessThan(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% lessThan_subset_iff
tff(fact_2918_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_2919_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_2920_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% sum.lessThan_Suc
tff(fact_2921_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),ring_1_of_int(A,Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),Z) ) ).

% ceiling_add_of_int
tff(fact_2922_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),ring_1_of_int(A,Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xb)),Z) ) ).

% ceiling_diff_of_int
tff(fact_2923_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_2924_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb) ) ) ).

% zero_less_ceiling
tff(fact_2925_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(num,A,numeral_numeral(A),V)) ) ) ).

% ceiling_le_numeral
tff(fact_2926_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_2927_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),V)),Xb) ) ) ).

% numeral_less_ceiling
tff(fact_2928_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb) ) ) ).

% one_le_ceiling
tff(fact_2929_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_2930_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ).

% one_less_ceiling
tff(fact_2931_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_2932_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_2933_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_2934_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_diff_numeral
tff(fact_2935_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xb)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_2936_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: num,Nb: nat] : archimedean_ceiling(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ).

% ceiling_numeral_power
tff(fact_2937_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),aa(num,num,bit0,Y))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_2938_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Xb))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_2939_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_dq(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2940_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_2941_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb) ) ) ).

% zero_le_ceiling
tff(fact_2942_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_2943_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_2944_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_2945_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_2946_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),Xb) ) ) ).

% numeral_le_ceiling
tff(fact_2947_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2948_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xb) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2949_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% and_Suc_0_eq
tff(fact_2950_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Suc_0_and_eq
tff(fact_2951_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_2952_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(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_2953_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),Xb) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2954_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_dr(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_2955_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,Xb)) ) ) ).

% ceiling_mono
tff(fact_2956_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),ring_1_of_int(A,archimedean_ceiling(A,Xb))) ) ).

% le_of_int_ceiling
tff(fact_2957_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% ceiling_less_cancel
tff(fact_2958_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Mb: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),Mb)),aa(A,set(A),set_ord_lessThan(A),Nb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb) ) ) ).

% lessThan_strict_subset_iff
tff(fact_2959_complex__mod__minus__le__complex__mod,axiom,
    ! [Xb: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Xb))),real_V7770717601297561774m_norm(complex,Xb)) ).

% complex_mod_minus_le_complex_mod
tff(fact_2960_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(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_2961_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),ring_1_of_int(A,A2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),A2) ) ) ).

% ceiling_le
tff(fact_2962_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),ring_1_of_int(A,Z)) ) ) ).

% ceiling_le_iff
tff(fact_2963_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,Z)),Xb) ) ) ).

% less_ceiling_iff
tff(fact_2964_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_2965_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Xb))),aa(real,real,exp(real),real_V7770717601297561774m_norm(A,Xb))) ) ).

% norm_exp
tff(fact_2966_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ds(fun(nat,A),fun(nat,fun(nat,A)),G),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.nat_diff_reindex
tff(fact_2967_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Nb: A] :
          ( ! [X2: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X2)),aa(A,nat,P,X2))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,P),aa(A,set(A),set_ord_lessThan(A),Nb))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Q),aa(A,set(A),set_ord_lessThan(A),Nb))) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_dt(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),Nb)) ) ) ) ).

% sum_diff_distrib
tff(fact_2968_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,archimedean_ceiling(A,R))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_2969_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,archimedean_ceiling(A,R))),one_one(A))),R) ) ).

% of_int_ceiling_diff_one_le
tff(fact_2970_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N4))),K6) )
        <=> ? [N5: nat] :
            ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N5))) ) ) ).

% lemma_NBseq_def2
tff(fact_2971_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N4))),K6) )
        <=> ? [N5: nat] :
            ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N5))) ) ) ).

% lemma_NBseq_def
tff(fact_2972_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),ring_1_of_int(A,A2)),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_2973_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = 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_cv(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% sum.lessThan_Suc_shift
tff(fact_2974_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_du(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,Mb)) ) ).

% sum_lessThan_telescope'
tff(fact_2975_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_db(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Mb)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_2976_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F2: fun(nat,A),Nb: nat,R: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),R)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_dv(fun(nat,A),fun(A,fun(nat,A)),F2),R)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sumr_diff_mult_const2
tff(fact_2977_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cv(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_2978_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,I4)),one_one(A))),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),ring_1_of_int(A,I4)) )
             => aa(int,$o,P,I4) ) ) ) ).

% ceiling_split
tff(fact_2979_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( archimedean_ceiling(A,Xb) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,A2)),one_one(A))),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),ring_1_of_int(A,A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_2980_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,Z)),one_one(A))),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),ring_1_of_int(A,Z))
           => ( archimedean_ceiling(A,Xb) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_2981_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,archimedean_ceiling(A,Xb))),one_one(A))),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),ring_1_of_int(A,archimedean_ceiling(A,Xb))) ) ) ).

% ceiling_correct
tff(fact_2982_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),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_2983_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_2984_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,Z)),one_one(A))),Xb) ) ) ).

% le_ceiling_iff
tff(fact_2985_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% power_diff_1_eq
tff(fact_2986_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% one_diff_power_eq
tff(fact_2987_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Nb: nat] :
          ( ( Xb != one_one(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_2988_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),times_times(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_2989_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => topological_monoseq(real,aa(real,fun(nat,real),power_power(real),Xb)) ) ) ).

% monoseq_realpow
tff(fact_2990_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ).

% sum_gp_strict
tff(fact_2991_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dw(A,fun(A,fun(nat,fun(nat,A))),Z),H),Mb)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dx(A,fun(A,fun(nat,fun(nat,A))),Z),H),Mb)),aa(nat,set(nat),set_ord_lessThan(nat),Mb)) ) ).

% lemma_termdiff1
tff(fact_2992_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dy(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb)))) ) ).

% diff_power_eq_sum
tff(fact_2993_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dz(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% power_diff_sumr2
tff(fact_2994_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(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_2995_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),ring_1_of_int(A,Nb)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,Nb)),one_one(A)))
           => ( archimedean_ceiling(A,Xb) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_2996_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,F2: fun(nat,A),K5: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,P4)),K5) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K5)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_2997_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% exp_bound_half
tff(fact_2998_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ea(A,fun(nat,fun(nat,A)),Xb),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% one_diff_power_eq'
tff(fact_2999_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Mb),Nb) = $ite(
        ( ( Mb = zero_zero(nat) )
        | ( Nb = zero_zero(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),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% and_nat_unfold
tff(fact_3000_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Mb),Nb) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% and_nat_rec
tff(fact_3001_sum__split__even__odd,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real),Nb: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_eb(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ec(fun(nat,real),fun(nat,real),F2)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ed(fun(nat,real),fun(nat,real),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ).

% sum_split_even_odd
tff(fact_3002_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))),real_V7770717601297561774m_norm(A,Z)))) ) ) ).

% exp_bound_lemma
tff(fact_3003_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Nb)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_3004_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Nb)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),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),Nb),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_3005_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,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,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_3006_arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,arctan,Xb) = suminf(real,aTP_Lamp_ee(real,fun(nat,real),Xb)) ) ) ).

% arctan_series
tff(fact_3007_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_3008_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_3009_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_3010_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_3011_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xb)),zero_zero(real))
        <=> ( Xb = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_3012_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_3013_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_3014_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xb))
        <=> ( Xb != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_3015_norm__minus__commute,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ).

% norm_minus_commute
tff(fact_3016_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_3017_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xb)) ) ).

% norm_ge_zero
tff(fact_3018_norm__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y)) ) ).

% norm_mult
tff(fact_3019_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_3020_sum__norm__le,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S: set(A),F2: fun(A,B),G: fun(A,real)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),S)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),aa(A,real,G,X2)) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S))),aa(set(A),real,groups7311177749621191930dd_sum(A,real,G),S)) ) ) ).

% sum_norm_le
tff(fact_3021_norm__power,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,Nb: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xb)),Nb) ) ).

% norm_power
tff(fact_3022_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),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_ef(fun(B,A),fun(B,real),F2)),A3)) ) ).

% norm_sum
tff(fact_3023_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xb)),Y)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ).

% norm_uminus_minus
tff(fact_3024_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_3025_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Nb: nat,Z: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( real_V7770717601297561774m_norm(A,W) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_3026_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A,R: real,Y: A,S3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R),S3)) ) ) ) ).

% norm_mult_less
tff(fact_3027_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_mult_ineq
tff(fact_3028_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,R: real,Y: A,S3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R),S3)) ) ) ) ).

% norm_add_less
tff(fact_3029_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E) ) ) ).

% norm_triangle_lt
tff(fact_3030_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C2)
         => aa(real,$o,aa(real,fun(real,$o),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_3031_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E) ) ) ).

% norm_triangle_le
tff(fact_3032_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_3033_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R: real,B2: A,S3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),S3)
           => aa(real,$o,aa(real,fun(real,$o),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),R),S3)) ) ) ) ).

% norm_triangle_mono
tff(fact_3034_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A,Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xb)),Nb)) ) ).

% norm_power_ineq
tff(fact_3035_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_3036_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)))) ) ).

% norm_triangle_sub
tff(fact_3037_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(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_3038_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_le
tff(fact_3039_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y))),E) ) ) ).

% norm_triangle_le_diff
tff(fact_3040_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(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_3041_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(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),minus_minus(A),A2),B2))) ) ).

% norm_triangle_ineq2
tff(fact_3042_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),Nb) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W) = one_one(real) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_3043_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(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),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% norm_diff_triangle_ineq
tff(fact_3044_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(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),minus_minus(A),A2),B2))) ) ).

% norm_triangle_ineq3
tff(fact_3045_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,Xb) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_3046_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,Mb: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,W)),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W),Mb)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Mb)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W)))) ) ) ) ).

% norm_power_diff
tff(fact_3047_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),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,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_3048_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_eg(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_3049_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_cv(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),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_3050_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_eh(nat,real)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_3051_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N2: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,N2)
         => ( ! [N: nat] :
                ( ~ aa(set(nat),$o,member(nat,N),N2)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( suminf(A,F2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N2) ) ) ) ) ).

% suminf_finite
tff(fact_3052_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3053_pi__series,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = suminf(real,aTP_Lamp_ei(nat,real)) ).

% pi_series
tff(fact_3054_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => summable(real,aTP_Lamp_ee(real,fun(nat,real),Xb)) ) ).

% summable_arctan_series
tff(fact_3055_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = Y )
     => ( Y = $ite(
            ( aa(set(int),$o,member(int,Xb),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,Xaa),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xb)
                & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xaa) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xb)
                  & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xaa) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),Xb),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.elims
tff(fact_3056_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( aa(set(int),$o,member(int,K),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & aa(set(int),$o,member(int,L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
            & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
        aa(int,int,
          aa(int,fun(int,int),plus_plus(int),
            aa($o,int,zero_neq_one_of_bool(int),
              ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
              & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
          aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ).

% and_int.simps
tff(fact_3057_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero(real) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,exp(real),Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ej(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3058_Diff__insert0,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B4: set(A)] :
      ( ~ aa(set(A),$o,member(A,Xb),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) ) ) ).

% Diff_insert0
tff(fact_3059_insert__Diff1,axiom,
    ! [A: $tType,Xb: A,B4: set(A),A3: set(A)] :
      ( aa(set(A),$o,member(A,Xb),B4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,Xb),A3)),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) ) ) ).

% insert_Diff1
tff(fact_3060_singleton__conv,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ek(A,fun(A,$o),A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_3061_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),fequal(A,A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_3062_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_el(nat,A)) ) ).

% summable_zero
tff(fact_3063_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I2: nat,F2: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_em(nat,fun(fun(nat,A),fun(nat,A)),I2),F2)) ) ).

% summable_single
tff(fact_3064_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(nat,fun(nat,A)),F2),K))
        <=> summable(A,F2) ) ) ).

% summable_iff_shift
tff(fact_3065_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = aa(set(A),set(A),insert(A,A2),A3) ).

% insert_Diff_single
tff(fact_3066_finite__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B4: set(A)] :
      ( finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)))
    <=> finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) ) ).

% finite_Diff_insert
tff(fact_3067_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3068_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_3069_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_eo(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_3070_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_ep(fun(nat,A),fun(A,fun(nat,A)),F2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_3071_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,A3)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eq(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F2)) ) ) ).

% summable_If_finite_set
tff(fact_3072_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( finite_finite2(nat,aa(fun(nat,$o),set(nat),collect(nat),P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_er(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_3073_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ~ aa(set(A),$o,member(A,Xb),A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,Xb),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_3074_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_3075_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Nb))),semiring_char_0_fact(A,Nb)) ) ).

% fact_Suc
tff(fact_3076_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,K),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K)) = aa(set(A),set(A),insert(A,K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_3077_fact__2,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).

% fact_2
tff(fact_3078_set__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,Nb,Xb)) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_3079_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,aa(A,fun(nat,A),power_power(A),C2))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_3080_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_es(fun(nat,A),fun(nat,real),F2))
         => summable(A,F2) ) ) ).

% summable_norm_cancel
tff(fact_3081_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] : aa(set(A),set(A),insert(A,A2),aa(fun(A,$o),set(A),collect(A),P)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_et(A,fun(fun(A,$o),fun(A,$o)),A2),P)) ).

% insert_Collect
tff(fact_3082_insert__compr,axiom,
    ! [A: $tType,A2: A,B4: set(A)] : aa(set(A),set(A),insert(A,A2),B4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_eu(A,fun(set(A),fun(A,$o)),A2),B4)) ).

% insert_compr
tff(fact_3083_insert__Diff__if,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B4: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,Xb),A3)),B4) = $ite(aa(set(A),$o,member(A,Xb),B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4),aa(set(A),set(A),insert(A,Xb),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))) ).

% insert_Diff_if
tff(fact_3084_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N2: nat,F2: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
           => summable(A,F2) ) ) ) ).

% summable_comparison_test'
tff(fact_3085_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N6: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test
tff(fact_3086_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_ev(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_3087_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_ew(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult2
tff(fact_3088_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_ex(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_mult
tff(fact_3089_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_ey(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_add
tff(fact_3090_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_ez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ).

% summable_diff
tff(fact_3091_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_fa(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_Suc_iff
tff(fact_3092_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_ep(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_divide
tff(fact_3093_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => summable(A,aTP_Lamp_fb(fun(nat,A),fun(nat,A),F2)) ) ) ).

% summable_minus
tff(fact_3094_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,aTP_Lamp_fb(fun(nat,A),fun(nat,A),F2))
        <=> summable(A,F2) ) ) ).

% summable_minus_iff
tff(fact_3095_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_en(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_3096_summable__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => summable(B,aa(A,fun(nat,B),F2,I3)) )
         => summable(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_fd(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) ) ) ).

% summable_sum
tff(fact_3097_summable__rabs__cancel,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_fe(fun(nat,real),fun(nat,real),F2))
     => summable(real,F2) ) ).

% summable_rabs_cancel
tff(fact_3098_Collect__conv__if,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ff(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_3099_Collect__conv__if2,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_fg(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_3100_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),F2),Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_fi(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_3101_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,G,N))
         => ( summable(A,F2)
           => ( summable(A,G)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),suminf(A,G)) ) ) ) ) ).

% suminf_le
tff(fact_3102_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_zero
tff(fact_3103_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Nb)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_3104_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_gt_zero
tff(fact_3105_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_1
tff(fact_3106_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Mb)),semiring_char_0_fact(A,Nb)) ) ) ).

% fact_mono
tff(fact_3107_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_eo(A,fun(fun(nat,A),fun(nat,A)),C2),F2))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_3108_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,Mb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,Mb)) ) ) ).

% fact_dvd
tff(fact_3109_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_3110_pi__not__less__zero,axiom,
    ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_3111_pi__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_3112_Compl__insert,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,Xb),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_3113_Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_3114_insert__Diff,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = A3 ) ) ).

% insert_Diff
tff(fact_3115_Diff__insert2,axiom,
    ! [A: $tType,A3: set(A),A2: A,B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))),B4) ).

% Diff_insert2
tff(fact_3116_Diff__insert__absorb,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] :
      ( ~ aa(set(A),$o,member(A,Xb),A3)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert(A,Xb),A3)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_3117_pi__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_3118_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),Xb: A,C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),aa(set(A),set(A),insert(A,Xb),C5)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),C5))
        & ~ aa(set(A),$o,member(A,Xb),A3) ) ) ).

% subset_Diff_insert
tff(fact_3119_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_ex(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_3120_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_ew(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ) ).

% suminf_mult2
tff(fact_3121_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_ey(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_add
tff(fact_3122_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,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)) ) ) ) ) ).

% suminf_diff
tff(fact_3123_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_ep(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_3124_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_fb(fun(nat,A),fun(nat,A),F2)) = aa(A,A,uminus_uminus(A),suminf(A,F2)) ) ) ) ).

% suminf_minus
tff(fact_3125_suminf__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B))] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => summable(B,aa(A,fun(nat,B),F2,I3)) )
         => ( suminf(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_fd(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_fj(fun(A,fun(nat,B)),fun(A,B),F2)),I5) ) ) ) ).

% suminf_sum
tff(fact_3126_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_nonneg
tff(fact_3127_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N4: nat] : aa(nat,A,F2,N4) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3128_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_pos
tff(fact_3129_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_fk(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_3130_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_fl(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_3131_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_fm(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_split_head
tff(fact_3132_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_fn(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_fo(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% powser_split_head(3)
tff(fact_3133_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Mb: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_fp(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F2),Mb),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_3134_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),P: fun(set(A),$o),F2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X2: A,S5: set(A)] :
                  ( finite_finite2(A,S5)
                 => ( ! [Y5: A] :
                        ( aa(set(A),$o,member(A,Y5),S5)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y5)),aa(A,B,F2,X2)) )
                   => ( aa(set(A),$o,P,S5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X2),S5)) ) ) )
             => aa(set(A),$o,P,S) ) ) ) ) ).

% finite_ranking_induct
tff(fact_3135_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B3: A,A7: set(A)] :
                  ( finite_finite2(A,A7)
                 => ( ! [X4: A] :
                        ( aa(set(A),$o,member(A,X4),A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),X4) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B3),A7)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3136_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B3: A,A7: set(A)] :
                  ( finite_finite2(A,A7)
                 => ( ! [X4: A] :
                        ( aa(set(A),$o,member(A,X4),A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),B3) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B3),A7)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3137_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Mb)),semiring_char_0_fact(A,Nb)) ) ) ) ).

% fact_less_mono
tff(fact_3138_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,Xb),A3)) = $ite(aa(set(A),$o,member(A,Xb),A3),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3))) ) ) ) ).

% sum.insert_if
tff(fact_3139_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),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,Nb))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))) ) ).

% fact_fact_dvd_fact
tff(fact_3140_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( modulo_modulo(A,semiring_char_0_fact(A,Nb),semiring_char_0_fact(A,Mb)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3141_infinite__remove,axiom,
    ! [A: $tType,S: set(A),A2: A] :
      ( ~ finite_finite2(A,S)
     => ~ finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_3142_infinite__coinduct,axiom,
    ! [A: $tType,X5: fun(set(A),$o),A3: set(A)] :
      ( aa(set(A),$o,X5,A3)
     => ( ! [A7: set(A)] :
            ( aa(set(A),$o,X5,A7)
           => ? [X4: A] :
                ( aa(set(A),$o,member(A,X4),A7)
                & ( aa(set(A),$o,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A)))))
                  | ~ finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A))))) ) ) )
       => ~ finite_finite2(A,A3) ) ) ).

% infinite_coinduct
tff(fact_3143_finite__empty__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),$o,P,A3)
       => ( ! [A4: A,A7: set(A)] :
              ( finite_finite2(A,A7)
             => ( aa(set(A),$o,member(A,A4),A7)
               => ( aa(set(A),$o,P,A7)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,A4),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_3144_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xb: A,B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xb),B4))
    <=> $ite(aa(set(A),$o,member(A,Xb),A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)) ) ).

% subset_insert_iff
tff(fact_3145_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),Xb: A,B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xb),B4)) ) ).

% Diff_single_insert
tff(fact_3146_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Nb)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),Nb))) ) ).

% fact_le_power
tff(fact_3147_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N6: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),aa(nat,real,G,N)) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_fq(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3148_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N6: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F2,N))),aa(nat,real,G,N)) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_fe(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_3149_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,Xb))),aa(set(A),set(A),insert(A,Xb),aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_3150_summable__rabs,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_fe(fun(nat,real),fun(nat,real),F2))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),suminf(real,F2))),suminf(real,aTP_Lamp_fe(fun(nat,real),fun(nat,real),F2))) ) ).

% summable_rabs
tff(fact_3151_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I2: nat] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ) ).

% suminf_pos2
tff(fact_3152_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2))
            <=> ? [I4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I4)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3153_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xb: A] :
          ( summable(A,F2)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),N))),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),Xb) ) ) ) ).

% suminf_le_const
tff(fact_3154_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(A,fun(nat,A)),F2),Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_3155_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xb: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),N))),Xb)
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3156_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),Xb)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3157_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),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_3158_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_fa(fun(nat,A),fun(nat,A),F2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3159_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$o,aa(A,fun(A,$o),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,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))),semiring_char_0_fact(A,Nb)) ) ) ).

% choose_dvd
tff(fact_3160_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),B4: set(A)] :
      ( aa(set(A),$o,P,bot_bot(set(A)))
     => ( ( ~ finite_finite2(A,B4)
         => aa(set(A),$o,P,B4) )
       => ( ! [A7: set(A)] :
              ( finite_finite2(A,A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B4)
                 => ( ! [X4: A] :
                        ( aa(set(A),$o,member(A,X4),A7)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B4) ) ) ) ).

% remove_induct
tff(fact_3161_finite__remove__induct,axiom,
    ! [A: $tType,B4: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,B4)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [A7: set(A)] :
              ( finite_finite2(A,A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B4)
                 => ( ! [X4: A] :
                        ( aa(set(A),$o,member(A,X4),A7)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,X4),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B4) ) ) ) ).

% finite_remove_induct
tff(fact_3162_pi__less__4,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) ).

% pi_less_4
tff(fact_3163_pi__ge__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi) ).

% pi_ge_two
tff(fact_3164_pi__half__neq__two,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)) ).

% pi_half_neq_two
tff(fact_3165_finite__induct__select,axiom,
    ! [A: $tType,S: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,S)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T5: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T5),S)
             => ( aa(set(A),$o,P,T5)
               => ? [X4: A] :
                    ( aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T5))
                    & aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),T5)) ) ) )
         => aa(set(A),$o,P,S) ) ) ) ).

% finite_induct_select
tff(fact_3166_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xb: A,B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),aa(set(A),set(A),insert(A,Xb),B4))
    <=> $ite(
          aa(set(A),$o,member(A,Xb),B4),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4),
          $ite(aa(set(A),$o,member(A,Xb),A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)) ) ) ).

% psubset_insert_iff
tff(fact_3167_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_3168_set__replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,Nb),Xb)) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_3169_set__replicate__conv__if,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      aa(list(A),set(A),set2(A),replicate(A,Nb,Xb)) = $ite(Nb = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_3170_sum__diff1__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A),A2: A] :
      aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A2),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(A,nat,F2,A2)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)) ).

% sum_diff1_nat
tff(fact_3171_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_es(fun(nat,A),fun(nat,real),F2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F2))),suminf(real,aTP_Lamp_es(fun(nat,A),fun(nat,real),F2))) ) ) ).

% summable_norm
tff(fact_3172_atLeastAtMostPlus1__int__conv,axiom,
    ! [Mb: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Mb),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb))
     => ( set_or1337092689740270186AtMost(int,Mb,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)) = aa(set(int),set(int),insert(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)),set_or1337092689740270186AtMost(int,Mb,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_3173_simp__from__to,axiom,
    ! [I2: int,J: int] :
      set_or1337092689740270186AtMost(int,I2,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I2),bot_bot(set(int)),aa(set(int),set(int),insert(int,I2),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J))) ).

% simp_from_to
tff(fact_3174_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I5: set(nat)] :
          ( summable(A,F2)
         => ( finite_finite2(nat,I5)
           => ( ! [N: nat] :
                  ( aa(set(nat),$o,member(nat,N),aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),I5)),suminf(A,F2)) ) ) ) ) ).

% sum_le_suminf
tff(fact_3175_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_en(fun(nat,A),fun(nat,fun(nat,A)),F2),K))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_3176_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_en(fun(nat,A),fun(nat,fun(nat,A)),F2),K)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_3177_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,Xb),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_3178_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,Xb),A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_3179_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A2),A3),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(A,B,F2,A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ) ).

% sum_diff1
tff(fact_3180_pi__half__neq__zero,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_3181_pi__half__less__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% pi_half_less_two
tff(fact_3182_pi__half__le__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% pi_half_le_two
tff(fact_3183_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_fr(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),S) = $ite(aa(set(A),$o,member(A,A2),S),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_3184_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat] :
          ( summable(A,F2)
         => ( ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,M)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),suminf(A,F2)) ) ) ) ).

% sum_less_suminf
tff(fact_3185_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,Nb)),semiring_char_0_fact(real,Nb))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))) ).

% square_fact_le_2_fact
tff(fact_3186_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_fn(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_fn(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_fo(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3187_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_fn(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_fo(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3188_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => ( summable(A,F2)
           => ? [N7: nat] :
              ! [N8: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N8)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(nat,fun(nat,A)),F2),N8)))),R) ) ) ) ) ).

% suminf_exist_split
tff(fact_3189_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E: real] :
          ( summable(A,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => ~ ! [N7: nat] :
                  ~ ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),M2)
                     => ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,M2,N8)))),E) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_3190_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,I3)),one_one(real))
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,I3))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Z)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_fs(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3191_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real,R0: real,A2: fun(nat,A),M6: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),R),R0)
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N))),M6)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_ft(real,fun(fun(nat,A),fun(nat,real)),R),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_3192_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I2: A,A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,I2),A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,I2),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
           => ( finite_finite2(A,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ) ) ).

% member_le_sum
tff(fact_3193_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Mb: nat] :
          semiring_char_0_fact(A,Mb) = $ite(Mb = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Mb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_3194_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( semiring_char_0_fact(A,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_3195_pi__half__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% pi_half_gt_zero
tff(fact_3196_pi__half__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))) ).

% pi_half_ge_zero
tff(fact_3197_m2pi__less__pi,axiom,
    aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))),pi))),pi) ).

% m2pi_less_pi
tff(fact_3198_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N2: nat,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,aa(nat,nat,suc,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N)))) )
           => summable(A,F2) ) ) ) ).

% summable_ratio_test
tff(fact_3199_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Nb: nat,I2: nat] :
          ( summable(A,F2)
         => ( ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),I2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),suminf(A,F2)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3200_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))) ).

% arctan_ubound
tff(fact_3201_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),aa(num,num,bit0,aa(num,num,bit0,one2)))) ).

% arctan_one
tff(fact_3202_minus__pi__half__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_3203_arctan__lbound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y)) ).

% arctan_lbound
tff(fact_3204_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))) ) ).

% arctan_bounded
tff(fact_3205_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: real,Nb: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xb = zero_zero(real) )
         => ( ( Nb != zero_zero(nat) )
           => ( aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_fu(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xb),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_3206_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),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_fv(real,fun(fun(nat,real),fun(nat,real)),H),J)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),Nb)),semiring_char_0_fact(real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_3207_Maclaurin__exp__le,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
      & ( aa(real,real,exp(real),Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ej(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_3208_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,aa(num,num,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),aa(num,num,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,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) ).

% machin_Euler
tff(fact_3209_machin,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,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,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,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))) ).

% machin
tff(fact_3210_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D3: nat] : aa(real,$o,aa(real,fun(real,$o),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)))),D3)))),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)))),D3)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,F2),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(real,F2)) ) ) ).

% sum_pos_lt_pair
tff(fact_3211_cos__coeff__def,axiom,
    ! [X4: nat] :
      cos_coeff(X4) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),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),aa(num,num,bit0,one2))))),semiring_char_0_fact(real,X4)),zero_zero(real)) ).

% cos_coeff_def
tff(fact_3212_sin__coeff__def,axiom,
    ! [X4: nat] :
      sin_coeff(X4) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X4),zero_zero(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))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X4),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),semiring_char_0_fact(real,X4))) ).

% sin_coeff_def
tff(fact_3213_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),zero_zero(real))
            & ( cos(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fw(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3214_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),Xb)
            & ( cos(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fw(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_3215_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
      & ( cos(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fw(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_3216_sin__cos__npi,axiom,
    ! [Nb: 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),aa(num,num,bit0,one2))),Nb)))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Nb) ).

% sin_cos_npi
tff(fact_3217_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_3218_sin__pi__minus,axiom,
    ! [Xb: real] : sin(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),Xb)) = sin(real,Xb) ).

% sin_pi_minus
tff(fact_3219_cos__pi,axiom,
    cos(real,pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_3220_cos__periodic__pi2,axiom,
    ! [Xb: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),Xb)) = aa(real,real,uminus_uminus(real),cos(real,Xb)) ).

% cos_periodic_pi2
tff(fact_3221_cos__periodic__pi,axiom,
    ! [Xb: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),cos(real,Xb)) ).

% cos_periodic_pi
tff(fact_3222_sin__periodic__pi2,axiom,
    ! [Xb: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),Xb)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ).

% sin_periodic_pi2
tff(fact_3223_sin__periodic__pi,axiom,
    ! [Xb: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ).

% sin_periodic_pi
tff(fact_3224_cos__minus__pi,axiom,
    ! [Xb: real] : cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),cos(real,Xb)) ).

% cos_minus_pi
tff(fact_3225_cos__pi__minus,axiom,
    ! [Xb: real] : cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),Xb)) = aa(real,real,uminus_uminus(real),cos(real,Xb)) ).

% cos_pi_minus
tff(fact_3226_sin__minus__pi,axiom,
    ! [Xb: real] : sin(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ).

% sin_minus_pi
tff(fact_3227_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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,Xb)),cos(A,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Xb))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_3228_sin__npi2,axiom,
    ! [Nb: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),Nb))) = zero_zero(real) ).

% sin_npi2
tff(fact_3229_sin__npi,axiom,
    ! [Nb: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = zero_zero(real) ).

% sin_npi
tff(fact_3230_sin__npi__int,axiom,
    ! [Nb: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),ring_1_of_int(real,Nb))) = zero_zero(real) ).

% sin_npi_int
tff(fact_3231_cos__pi__half,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_3232_sin__two__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) = zero_zero(real) ).

% sin_two_pi
tff(fact_3233_sin__pi__half,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_3234_cos__two__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) = one_one(real) ).

% cos_two_pi
tff(fact_3235_cos__periodic,axiom,
    ! [Xb: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = cos(real,Xb) ).

% cos_periodic
tff(fact_3236_sin__periodic,axiom,
    ! [Xb: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = sin(real,Xb) ).

% sin_periodic
tff(fact_3237_cos__2pi__minus,axiom,
    ! [Xb: real] : cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),Xb)) = cos(real,Xb) ).

% cos_2pi_minus
tff(fact_3238_cos__npi,axiom,
    ! [Nb: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Nb) ).

% cos_npi
tff(fact_3239_cos__npi2,axiom,
    ! [Nb: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Nb) ).

% cos_npi2
tff(fact_3240_set__encode__insert,axiom,
    ! [A3: set(nat),Nb: nat] :
      ( finite_finite2(nat,A3)
     => ( ~ aa(set(nat),$o,member(nat,Nb),A3)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,Nb),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),aa(num,num,bit0,one2))),Nb)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_3241_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_3242_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_3243_sin__2npi,axiom,
    ! [Nb: 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),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_3244_cos__2npi,axiom,
    ! [Nb: 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),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_3245_sin__2pi__minus,axiom,
    ! [Xb: real] : sin(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),Xb)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ).

% sin_2pi_minus
tff(fact_3246_sin__int__2pin,axiom,
    ! [Nb: 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),aa(num,num,bit0,one2))),pi)),ring_1_of_int(real,Nb))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_3247_cos__int__2pin,axiom,
    ! [Nb: 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),aa(num,num,bit0,one2))),pi)),ring_1_of_int(real,Nb))) = one_one(real) ).

% cos_int_2pin
tff(fact_3248_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),aa(num,num,bit0,one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_3249_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),aa(num,num,bit0,one2)))),pi)) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% sin_3over2_pi
tff(fact_3250_cos__npi__int,axiom,
    ! [Nb: int] :
      cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),ring_1_of_int(real,Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ).

% cos_npi_int
tff(fact_3251_cos__pi__eq__zero,axiom,
    ! [Mb: 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),aa(num,num,bit0,one2))),Mb))))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_3252_fact__ge__self,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),semiring_char_0_fact(nat,Nb)) ).

% fact_ge_self
tff(fact_3253_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),semiring_char_0_fact(nat,Mb)),semiring_char_0_fact(nat,Nb)) ) ).

% fact_mono_nat
tff(fact_3254_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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,Xb)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),sin(A,Y))) ) ).

% sin_add
tff(fact_3255_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) = one_one(A) )
         => ( sin(A,Xb) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_3256_polar__Ex,axiom,
    ! [Xb: real,Y: real] :
    ? [R3: real,A4: real] :
      ( ( Xb = 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_3257_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),sin(A,Y))) ) ).

% sin_diff
tff(fact_3258_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fx(set(A),fun(A,$o),A3)) ).

% Compl_eq
tff(fact_3259_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fy(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P)) ).

% Collect_neg_eq
tff(fact_3260_uminus__set__def,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3))) ).

% uminus_set_def
tff(fact_3261_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% cos_add
tff(fact_3262_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),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,Xb)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% cos_diff
tff(fact_3263_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,Xb)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_3264_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),cos(real,Xb)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_3265_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb)) = 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),aa(num,num,bit0,one2))),sin(A,Xb))),cos(A,Xb)) ) ).

% sin_double
tff(fact_3266_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y4)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),pi)
      & ( sin(real,Y4) = sin(real,Xb) )
      & ( cos(real,Y4) = cos(real,Xb) ) ) ).

% sincos_principal_value
tff(fact_3267_fact__less__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,Mb)),semiring_char_0_fact(nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_3268_sin__x__le__x,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),Xb) ) ).

% sin_x_le_x
tff(fact_3269_sin__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),one_one(real)) ).

% sin_le_one
tff(fact_3270_cos__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xb)),one_one(real)) ).

% cos_le_one
tff(fact_3271_abs__sin__x__le__abs__x,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xb))),aa(real,real,abs_abs(real),Xb)) ).

% abs_sin_x_le_abs_x
tff(fact_3272_lessThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,K),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% lessThan_Suc
tff(fact_3273_sin__cos__le1,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),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,Xb)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,Xb)),cos(real,Y))))),one_one(real)) ).

% sin_cos_le1
tff(fact_3274_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% sin_squared_eq
tff(fact_3275_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cos_squared_eq
tff(fact_3276_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_3277_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Mb),semiring_char_0_fact(nat,Nb)) ) ) ).

% dvd_fact
tff(fact_3278_sin__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero
tff(fact_3279_sin__x__ge__neg__x,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xb)),sin(real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_3280_sin__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_ge_zero
tff(fact_3281_sin__ge__minus__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),sin(real,Xb)) ).

% sin_ge_minus_one
tff(fact_3282_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xb)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_3283_cos__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xb)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_3284_cos__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( ( cos(real,Xb) = cos(real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_3285_cos__ge__minus__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),cos(real,Xb)) ).

% cos_ge_minus_one
tff(fact_3286_abs__sin__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xb))),one_one(real)) ).

% abs_sin_le_one
tff(fact_3287_abs__cos__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),cos(real,Xb))),one_one(real)) ).

% abs_cos_le_one
tff(fact_3288_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,aa(A,fun(A,A),minus_minus(A),cos(A,aa(A,A,aa(A,fun(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),aa(num,num,bit0,one2))) ) ).

% sin_times_sin
tff(fact_3289_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,aa(A,fun(A,A),minus_minus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sin_times_cos
tff(fact_3290_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,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),W),Z)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% cos_times_sin
tff(fact_3291_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),aa(num,num,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),aa(num,num,bit0,one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% sin_plus_sin
tff(fact_3292_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(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),aa(num,num,bit0,one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))))) ) ).

% sin_diff_sin
tff(fact_3293_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(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),aa(num,num,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),aa(num,num,bit0,one2)))))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% cos_diff_cos
tff(fact_3294_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cos_double
tff(fact_3295_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_3296_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(set(nat),set(nat),insert(nat,Mb),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) = set_or1337092689740270186AtMost(nat,Mb,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_3297_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
     => ( set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3298_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( set_or1337092689740270186AtMost(nat,Mb,Nb) = aa(set(nat),set(nat),insert(nat,Mb),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3299_lessThan__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_lessThan(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_3300_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),aa(num,num,bit0,one2))),W)) = aa(A,A,aa(A,fun(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),aa(num,num,bit0,one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,W)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% cos_double_sin
tff(fact_3301_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Mb))
     => ( semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Mb)),Nb)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_3302_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_3303_fact__div__fact__le__pow,axiom,
    ! [R: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Nb)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),R)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R)) ) ).

% fact_div_fact_le_pow
tff(fact_3304_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xb)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3305_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xb)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3306_sin__eq__0__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),pi)
       => ( ( sin(real,Xb) = zero_zero(real) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3307_sin__zero__pi__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),pi)
     => ( ( sin(real,Xb) = zero_zero(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3308_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Y)),cos(real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_3309_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I4: int] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,I4)),pi) ) ).

% sin_zero_iff_int2
tff(fact_3310_sincos__total__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),pi)
            & ( Xb = cos(real,T4) )
            & ( Y = sin(real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_3311_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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,Mb))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),Mb)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% sin_expansion_lemma
tff(fact_3312_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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,Mb))),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = aa(real,real,uminus_uminus(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),Mb)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% cos_expansion_lemma
tff(fact_3313_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_3314_cos__two__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),zero_zero(real)) ).

% cos_two_less_zero
tff(fact_3315_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
      & ( cos(real,X2) = zero_zero(real) )
      & ! [Y5: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
            & ( cos(real,Y5) = zero_zero(real) ) )
         => ( Y5 = X2 ) ) ) ).

% cos_is_zero
tff(fact_3316_cos__two__le__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_3317_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Y)),cos(real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3318_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),pi)
            & ( cos(real,X2) = Y )
            & ! [Y5: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),pi)
                  & ( cos(real,Y5) = Y ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_3319_sincos__total__pi__half,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
         => ? [T4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
              & ( Xb = cos(real,T4) )
              & ( Y = sin(real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_3320_sincos__total__2pi__le,axiom,
    ! [Xb: 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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))
          & ( Xb = cos(real,T4) )
          & ( Y = sin(real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_3321_sincos__total__2pi,axiom,
    ! [Xb: 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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
     => ~ ! [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))
             => ( ( Xb = cos(real,T4) )
               => ( Y != sin(real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3322_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(real,$o,aa(real,fun(real,$o),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),Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_3323_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,aa(A,fun(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),aa(num,num,bit0,one2))) ) ).

% cos_times_cos
tff(fact_3324_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),aa(num,num,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),aa(num,num,bit0,one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),W),Z)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% cos_plus_cos
tff(fact_3325_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa(set(nat),$o,member(nat,Nb),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),aa(num,num,bit0,one2))),Nb)),Z)) = aa(set(nat),set(nat),insert(nat,Nb),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_3326_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_3327_sin__lt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_3328_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Xb))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_3329_sin__30,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,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),aa(num,num,bit0,one2))) ).

% sin_30
tff(fact_3330_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_3331_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( ( sin(real,Xb) = sin(real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3332_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3333_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3334_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),aa(num,num,bit0,one2))) ).

% cos_60
tff(fact_3335_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = one_one(real) )
    <=> ? [X: int] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3336_sin__coeff__Suc,axiom,
    ! [Nb: nat] : sin_coeff(aa(nat,nat,suc,Nb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos_coeff(Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb))) ).

% sin_coeff_Suc
tff(fact_3337_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),aa(num,num,bit0,one2))),W)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,W)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_3338_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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))),Xb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),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,Xb))) ) ).

% cos_treble_cos
tff(fact_3339_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),pi),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_3340_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_3341_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xb)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3342_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3343_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),X2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
            & ( sin(real,X2) = Y )
            & ! [Y5: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
                  & ( sin(real,Y5) = Y ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_3344_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_3345_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_3346_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = one_one(real) )
    <=> ( ? [X: nat] : Xb = 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),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)
        | ? [X: nat] : Xb = 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),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_3347_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] : sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fz(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_3348_cos__coeff__Suc,axiom,
    ! [Nb: nat] : cos_coeff(aa(nat,nat,suc,Nb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),sin_coeff(Nb))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb))) ).

% cos_coeff_Suc
tff(fact_3349_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => aa(real,$o,aa(real,fun(real,$o),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),Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_3350_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
      & ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fz(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_3351_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I4: int] :
          ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),I4)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_3352_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),Xb)
          & ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fz(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_3353_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = zero_zero(real) )
    <=> ? [I4: int] :
          ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),I4)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_3354_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),Xb)
            & ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fz(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_3355_sin__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( sin(real,Xb) = zero_zero(real) )
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_3356_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ( ? [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4)
            & ( Xb = 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),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_3357_cos__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( cos(real,Xb) = zero_zero(real) )
       => ? [N: nat] :
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_3358_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = zero_zero(real) )
    <=> ( ? [N4: nat] :
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N4: nat] :
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4)
            & ( Xb = 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),N4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_3359_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb)) != 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),aa(num,num,bit0,one2))),Xb)) = 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),aa(num,num,bit0,one2))),aa(A,A,tan(A),Xb))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ) ).

% tan_double
tff(fact_3360_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))
             => ( Z != complex2(cos(real,T4),sin(real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3361_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = 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),Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( aa(set(int),$o,member(int,Xb),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & aa(set(int),$o,member(int,Xaa),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xb)
                      & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xaa) ))),
                  aa(int,int,
                    aa(int,fun(int,int),plus_plus(int),
                      aa($o,int,zero_neq_one_of_bool(int),
                        ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xb)
                        & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Xaa) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),Xb),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),aa(num,num,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),Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_3362_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))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
            ( aa(set(int),$o,member(int,K),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & aa(set(int),$o,member(int,L),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
                & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
                  & ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_3363_sin__paired,axiom,
    ! [Xb: real] : aa(real,$o,sums(real,aTP_Lamp_ga(real,fun(nat,real),Xb)),sin(real,Xb)) ).

% sin_paired
tff(fact_3364_tan__periodic__pi,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),pi)) = aa(real,real,tan(real),Xb) ).

% tan_periodic_pi
tff(fact_3365_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => aa(A,$o,sums(A,aTP_Lamp_el(nat,A)),zero_zero(A)) ) ).

% sums_zero
tff(fact_3366_tan__npi,axiom,
    ! [Nb: 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),Nb)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_3367_tan__periodic__n,axiom,
    ! [Xb: real,Nb: num] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),Nb)),pi))) = aa(real,real,tan(real),Xb) ).

% tan_periodic_n
tff(fact_3368_tan__periodic__nat,axiom,
    ! [Xb: real,Nb: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi))) = aa(real,real,tan(real),Xb) ).

% tan_periodic_nat
tff(fact_3369_tan__periodic__int,axiom,
    ! [Xb: real,I2: int] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,I2)),pi))) = aa(real,real,tan(real),Xb) ).

% tan_periodic_int
tff(fact_3370_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xb: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_fl(fun(nat,A),fun(nat,A),A2)),Xb)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xb ) ) ) ).

% powser_sums_zero_iff
tff(fact_3371_norm__cos__sin,axiom,
    ! [Ta: real] : real_V7770717601297561774m_norm(complex,complex2(cos(real,Ta),sin(real,Ta))) = one_one(real) ).

% norm_cos_sin
tff(fact_3372_tan__periodic,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = aa(real,real,tan(real),Xb) ).

% tan_periodic
tff(fact_3373_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A),S3: A,Ta: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,G,N))
         => ( aa(A,$o,sums(A,F2),S3)
           => ( aa(A,$o,sums(A,G),Ta)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S3),Ta) ) ) ) ) ).

% sums_le
tff(fact_3374_complex__diff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),complex2(A2,B2)),complex2(C2,D2)) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C2),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),D2)) ).

% complex_diff
tff(fact_3375_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I2: nat,F2: fun(nat,A)] : aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_em(nat,fun(fun(nat,A),fun(nat,A)),I2),F2)),aa(nat,A,F2,I2)) ) ).

% sums_single
tff(fact_3376_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ew(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_3377_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ex(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_3378_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ey(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_3379_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% sums_diff
tff(fact_3380_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ep(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_3381_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,F2),A2)
         => aa(A,$o,sums(A,aTP_Lamp_fb(fun(nat,A),fun(nat,A),F2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% sums_minus
tff(fact_3382_sums__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F2: fun(A,fun(nat,B)),Xb: fun(A,B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => aa(B,$o,sums(B,aa(A,fun(nat,B),F2,I3)),aa(A,B,Xb,I3)) )
         => aa(B,$o,sums(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_fd(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xb),I5)) ) ) ).

% sums_sum
tff(fact_3383_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gb(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),aa(A,A,aa(A,fun(A,A),times_times(A),D2),C2))
          <=> aa(A,$o,sums(A,F2),D2) ) ) ) ).

% sums_mult2_iff
tff(fact_3384_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F2: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gc(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))
          <=> aa(A,$o,sums(A,F2),D2) ) ) ) ).

% sums_mult_iff
tff(fact_3385_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_3386_complex__add,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),complex2(A2,B2)),complex2(C2,D2)) = 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),D2)) ).

% complex_add
tff(fact_3387_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F2: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eo(A,fun(fun(nat,A),fun(nat,A)),C2),F2)),A2)
         => ( ( C2 != zero_zero(A) )
           => aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% sums_mult_D
tff(fact_3388_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S3: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( aa(A,$o,sums(A,aTP_Lamp_fa(fun(nat,A),fun(nat,A),F2)),S3)
           => aa(A,$o,sums(A,F2),S3) ) ) ) ).

% sums_Suc_imp
tff(fact_3389_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S3: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_fa(fun(nat,A),fun(nat,A),F2)),S3)
        <=> aa(A,$o,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_3390_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_gd(fun(nat,A),fun(nat,A),F2)),L)
         => aa(A,$o,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_3391_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Nb: nat,F2: fun(nat,A),S3: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
             => ( aa(nat,A,F2,I3) = zero_zero(A) ) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ge(nat,fun(fun(nat,A),fun(nat,A)),Nb),F2)),S3)
          <=> aa(A,$o,sums(A,F2),S3) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3392_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_3393_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,A3)
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eq(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_3394_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( finite_finite2(nat,aa(fun(nat,$o),set(nat),collect(nat),P))
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_er(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(fun(nat,$o),set(nat),collect(nat),P))) ) ) ).

% sums_If_finite
tff(fact_3395_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N2: set(nat),F2: fun(nat,A)] :
          ( finite_finite2(nat,N2)
         => ( ! [N: nat] :
                ( ~ aa(set(nat),$o,member(nat,N),N2)
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => aa(A,$o,sums(A,F2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),N2)) ) ) ) ).

% sums_finite
tff(fact_3396_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),complex2(C2,D2)) = complex2(aa(real,real,aa(real,fun(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),D2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),D2)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),C2))) ).

% complex_mult
tff(fact_3397_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_3398_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_3399_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Mb: nat,Z: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_gf(nat,fun(A,fun(nat,A)),Mb),Z)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Mb)) ) ).

% powser_sums_if
tff(fact_3400_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : aa(A,$o,sums(A,aTP_Lamp_fl(fun(nat,A),fun(nat,A),A2)),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3401_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_3402_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S3: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb)),S3)
        <=> aa(A,$o,sums(A,F2),aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))) ) ) ).

% sums_iff_shift
tff(fact_3403_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S3: A,Nb: nat] :
          ( aa(A,$o,sums(A,F2),S3)
         => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))) ) ) ).

% sums_split_initial_segment
tff(fact_3404_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),Nb: nat,S3: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(nat,fun(nat,A)),F2),Nb)),aa(A,A,aa(A,fun(A,A),minus_minus(A),S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_lessThan(nat),Nb))))
        <=> aa(A,$o,sums(A,F2),S3) ) ) ).

% sums_iff_shift'
tff(fact_3405_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S: A,A3: set(nat),S4: A,F2: fun(nat,A)] :
          ( aa(A,$o,sums(A,G),S)
         => ( finite_finite2(nat,A3)
           => ( ( S4 = 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_gg(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A3)) )
             => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gh(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F2)),S4) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_3406_Complex__sum_H,axiom,
    ! [A: $tType,F2: fun(A,real),S3: set(A)] : aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_gi(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_3407_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_3408_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),aa(num,num,bit0,aa(num,num,bit0,one2))))) = one_one(real) ).

% tan_45
tff(fact_3409_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => aa(A,$o,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,aa(A,fun(A,A),minus_minus(A),one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_3410_power__half__series,axiom,
    aa(real,$o,sums(real,aTP_Lamp_gj(nat,real)),one_one(real)) ).

% power_half_series
tff(fact_3411_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tan(real),Xb)) ) ) ).

% tan_gt_zero
tff(fact_3412_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,tan(real),X2)) ) ) ).

% lemma_tan_total
tff(fact_3413_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),X2)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( aa(real,real,tan(real),X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_3414_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3415_tan__monotone_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3416_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xb)) ) ) ) ).

% tan_monotone
tff(fact_3417_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),X2)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( aa(real,real,tan(real),X2) = Y )
      & ! [Y5: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
            & ( aa(real,real,tan(real),Y5) = Y ) )
         => ( Y5 = X2 ) ) ) ).

% tan_total
tff(fact_3418_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),aa(num,num,bit0,aa(num,num,bit0,one2)))))) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% tan_minus_45
tff(fact_3419_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,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y)) ).

% tan_inverse
tff(fact_3420_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != 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),Xb)),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),Xb),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3421_sums__if_H,axiom,
    ! [G: fun(nat,real),Xb: real] :
      ( aa(real,$o,sums(real,G),Xb)
     => aa(real,$o,sums(real,aTP_Lamp_gk(fun(nat,real),fun(nat,real),G)),Xb) ) ).

% sums_if'
tff(fact_3422_sums__if,axiom,
    ! [G: fun(nat,real),Xb: real,F2: fun(nat,real),Y: real] :
      ( aa(real,$o,sums(real,G),Xb)
     => ( aa(real,$o,sums(real,F2),Y)
       => aa(real,$o,sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_gl(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% sums_if
tff(fact_3423_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
          & ( aa(real,real,tan(real),X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_3424_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_3425_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xb)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3426_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3427_tan__mono__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_3428_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),Xb))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3429_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( ( aa(real,real,tan(real),Xb) = Y )
         => ( aa(real,real,arctan,Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_3430_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_3431_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))
      & ( aa(real,real,tan(real),aa(real,real,arctan,Y)) = Y ) ) ).

% arctan
tff(fact_3432_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K2: int,L3: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L3))
           => ( ( ~ ( aa(set(int),$o,member(int,K2),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & aa(set(int),$o,member(int,L3),aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L3),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K2),L3) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_3433_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),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),Xb),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3434_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,tan(A),Xb)),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),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3435_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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),Xb)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3436_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ? [Z4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,aa(num,num,bit0,one2)))))),Z4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))
          & ( aa(real,real,tan(real),Z4) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_3437_cos__paired,axiom,
    ! [Xb: real] : aa(real,$o,sums(real,aTP_Lamp_gm(real,fun(nat,real),Xb)),cos(real,Xb)) ).

% cos_paired
tff(fact_3438_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tan(A),Xb) = 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),aa(num,num,bit0,one2))),Xb))),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),aa(num,num,bit0,one2))),Xb))),one_one(A))) ) ).

% tan_half
tff(fact_3439_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => aa(A,$o,sums(A,aTP_Lamp_gn(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,aa(A,fun(A,A),minus_minus(A),one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_3440_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I3: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I3),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_3441_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_gp(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),suminf(A,aa(A,fun(nat,A),aTP_Lamp_go(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))) ) ) ).

% diffs_equiv
tff(fact_3442_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),Xb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),Xb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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_3443_sin__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,tan(real),Xb)),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),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% sin_tan
tff(fact_3444_cos__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => ( cos(real,Xb) = 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),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% cos_tan
tff(fact_3445_real__sqrt__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,real,sqrt,Xb) = aa(real,real,sqrt,Y) )
    <=> ( Xb = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_3446_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_3447_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,sqrt,Xb) = zero_zero(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_3448_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_3449_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% real_sqrt_less_iff
tff(fact_3450_real__sqrt__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% real_sqrt_le_iff
tff(fact_3451_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_3452_real__sqrt__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,sqrt,Xb) = one_one(real) )
    <=> ( Xb = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_3453_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] :
          powr(A,Xb,zero_zero(A)) = $ite(Xb = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_3454_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_3455_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_3456_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_3457_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_3458_powr__gt__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),powr(real,Xb,A2))
    <=> ( Xb != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_3459_powr__nonneg__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,A2,Xb)),zero_zero(real))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_3460_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_3461_real__sqrt__lt__1__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_3462_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_3463_real__sqrt__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_3464_powr__less__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Xb,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_3465_real__sqrt__abs2,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Xb)) = aa(real,real,abs_abs(real),Xb) ).

% real_sqrt_abs2
tff(fact_3466_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_3467_real__sqrt__four,axiom,
    aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)) ).

% real_sqrt_four
tff(fact_3468_powr__eq__one__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( ( powr(real,A2,Xb) = one_one(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_3469_powr__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,one_one(real)) = Xb ) ) ).

% powr_one
tff(fact_3470_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( powr(real,Xb,one_one(real)) = Xb )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_3471_powr__le__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Xb,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_3472_numeral__powr__numeral__real,axiom,
    ! [Mb: num,Nb: num] : powr(real,aa(num,real,numeral_numeral(real),Mb),aa(num,real,numeral_numeral(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),Mb)),aa(num,nat,numeral_numeral(nat),Nb)) ).

% numeral_powr_numeral_real
tff(fact_3473_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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_3474_powr__log__cancel,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( powr(real,A2,aa(real,real,log(A2),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_3475_real__sqrt__abs,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(real,real,abs_abs(real),Xb) ).

% real_sqrt_abs
tff(fact_3476_powr__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(num,real,numeral_numeral(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),Nb)) ) ) ).

% powr_numeral
tff(fact_3477_real__sqrt__pow2__iff,axiom,
    ! [Xb: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Xb )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% real_sqrt_pow2_iff
tff(fact_3478_real__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Xb ) ) ).

% real_sqrt_pow2
tff(fact_3479_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Y: real,Xaa: 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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_3480_square__powr__half,axiom,
    ! [Xb: real] : powr(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(real,real,abs_abs(real),Xb) ).

% square_powr_half
tff(fact_3481_real__sqrt__minus,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,Xb)) ).

% real_sqrt_minus
tff(fact_3482_real__sqrt__power,axiom,
    ! [Xb: real,K: nat] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xb)),K) ).

% real_sqrt_power
tff(fact_3483_real__sqrt__mult,axiom,
    ! [Xb: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ).

% real_sqrt_mult
tff(fact_3484_real__sqrt__divide,axiom,
    ! [Xb: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ).

% real_sqrt_divide
tff(fact_3485_powr__powr,axiom,
    ! [Xb: real,A2: real,B2: real] : powr(real,powr(real,Xb,A2),B2) = powr(real,Xb,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)) ).

% powr_powr
tff(fact_3486_real__sqrt__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_3487_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_3488_powr__non__neg,axiom,
    ! [A2: real,Xb: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,A2,Xb)),zero_zero(real)) ).

% powr_non_neg
tff(fact_3489_powr__less__mono2__neg,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Y,A2)),powr(real,Xb,A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_3490_powr__mono2,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mono2
tff(fact_3491_powr__ge__pzero,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),powr(real,Xb,Y)) ).

% powr_ge_pzero
tff(fact_3492_real__sqrt__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_3493_real__sqrt__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_3494_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( aa(real,real,sqrt,Xb) = zero_zero(real) )
       => ( Xb = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_3495_powr__less__mono,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Xb,B2)) ) ) ).

% powr_less_mono
tff(fact_3496_powr__less__cancel,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Xb,B2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel
tff(fact_3497_powr__mono,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Xb,B2)) ) ) ).

% powr_mono
tff(fact_3498_real__sqrt__ge__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_ge_one
tff(fact_3499_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X4: nat] : aa(nat,A,diffs(A,aTP_Lamp_gq(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_3500_powr__less__mono2,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xb,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_less_mono2
tff(fact_3501_powr__mono2_H,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Y,A2)),powr(real,Xb,A2)) ) ) ) ).

% powr_mono2'
tff(fact_3502_gr__one__powr,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),powr(real,Xb,Y)) ) ) ).

% gr_one_powr
tff(fact_3503_powr__inj,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,Xb) = powr(real,A2,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_3504_powr__le1,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_3505_powr__mono__both,axiom,
    ! [A2: real,B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Y,B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_3506_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),powr(real,Xb,A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_3507_powr__divide,axiom,
    ! [Xb: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xb,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_divide
tff(fact_3508_powr__mult,axiom,
    ! [Xb: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Xb,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mult
tff(fact_3509_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,sqrt,Xb)) = aa(real,real,sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_3510_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_3511_le__real__sqrt__sumsq,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),Xb),Xb)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_3512_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_3513_log__powr,axiom,
    ! [Xb: real,B2: real,Y: real] :
      ( ( Xb != zero_zero(real) )
     => ( aa(real,real,log(B2),powr(real,Xb,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,log(B2),Xb)) ) ) ).

% log_powr
tff(fact_3514_ln__powr,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != zero_zero(real) )
     => ( aa(real,real,ln_ln(real),powr(real,Xb,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_powr
tff(fact_3515_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A2: A,B2: A] : powr(A,Xb,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,Xb,A2)),powr(A,Xb,B2)) ) ).

% powr_add
tff(fact_3516_powr__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [W: A,Z1: A,Z22: A] : powr(A,W,aa(A,A,aa(A,fun(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_3517_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(real,real,sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_3518_sqrt2__less__2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sqrt2_less_2
tff(fact_3519_powr__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb) ) ) ).

% powr_realpow
tff(fact_3520_powr__less__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(B2),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_3521_less__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_3522_log__less__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),Xb)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,Y)) ) ) ) ).

% log_less_iff
tff(fact_3523_less__log__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(B2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_3524_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_3525_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A2: A] : powr(A,Xb,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),powr(A,Xb,A2)) ) ).

% powr_minus_divide
tff(fact_3526_real__less__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_3527_real__le__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_3528_sqrt__le__D,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% sqrt_le_D
tff(fact_3529_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xb) ) ) ).

% powr_neg_one
tff(fact_3530_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [X2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),X2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)) ) ) ).

% termdiff_converges_all
tff(fact_3531_powr__mult__base,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,Xb,Y)) = powr(real,Xb,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_3532_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_3533_powr__le__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(B2),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_3534_le__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_3535_log__le__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),Xb)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,Y)) ) ) ) ).

% log_le_iff
tff(fact_3536_le__log__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(B2),Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_3537_real__sqrt__unique,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Xb )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,real,sqrt,Xb) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_3538_real__le__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_3539_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),U)
     => aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_3540_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: 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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = Y )
     => ( Xb = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_3541_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: 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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = Xb )
     => ( Y = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_3542_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,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),D2)),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_3543_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xb: real] : aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_3544_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_3545_sqrt__ge__absD,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,sqrt,Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y) ) ).

% sqrt_ge_absD
tff(fact_3546_ln__powr__bound,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xb,A2)),A2)) ) ) ).

% ln_powr_bound
tff(fact_3547_ln__powr__bound2,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),Xb),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_3548_cos__45,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,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,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_45
tff(fact_3549_sin__45,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,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,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_45
tff(fact_3550_add__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(B2),Xb)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_3551_log__add__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B2),Xb)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_3552_minus__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),aa(real,real,log(B2),Xb)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B2,Y)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_3553_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A,A2: A] :
          powr(A,Xb,A2) = $ite(Xb = zero_zero(A),zero_zero(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),Xb)))) ) ).

% powr_def
tff(fact_3554_real__less__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xb)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_3555_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_3556_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),aa(num,num,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_3557_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_3558_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% real_sqrt_ge_abs1
tff(fact_3559_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xb: real] : aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% real_sqrt_ge_abs2
tff(fact_3560_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,abs_abs(real),Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_3561_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xb)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% ln_sqrt
tff(fact_3562_cos__30,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,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),aa(num,num,bit0,one2))) ).

% cos_30
tff(fact_3563_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),aa(num,num,bit0,one2))) ).

% sin_60
tff(fact_3564_log__minus__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(B2),Xb)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_3565_complex__norm,axiom,
    ! [Xb: real,Y: real] : real_V7770717601297561774m_norm(complex,complex2(Xb,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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% complex_norm
tff(fact_3566_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,K5: real,C2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),K5)
         => ( ! [X2: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X2)),K5)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),X2)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gt(A,fun(fun(nat,A),fun(nat,A)),Xb),C2)) ) ) ) ).

% termdiff_converges
tff(fact_3567_arsinh__real__aux,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_3568_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xb)),Nb) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_3569_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_3570_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_3571_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),Nb))) = 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),Xb),aa(num,nat,numeral_numeral(nat),Nb))) ) ) ).

% powr_neg_numeral
tff(fact_3572_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),one_one(real)) ).

% cos_x_y_le_one
tff(fact_3573_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))))
       => aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_3574_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( aa(real,real,arcosh(real),Xb) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_3575_cos__arctan,axiom,
    ! [Xb: real] : cos(real,aa(real,real,arctan,Xb)) = 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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% cos_arctan
tff(fact_3576_sin__arctan,axiom,
    ! [Xb: real] : sin(real,aa(real,real,arctan,Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% sin_arctan
tff(fact_3577_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
           => aa(real,$o,aa(real,fun(real,$o),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_3578_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xb))
     => ( sin(real,Xb) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3579_arctan__half,axiom,
    ! [Xb: real] : aa(real,real,arctan,Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))) ).

% arctan_half
tff(fact_3580_arsinh__real__def,axiom,
    ! [Xb: real] : aa(real,real,arsinh(real),Xb) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))))) ).

% arsinh_real_def
tff(fact_3581_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] : aa(A,A,arcosh(A),Xb) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),powr(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))))))) ) ).

% arcosh_def
tff(fact_3582_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xb)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3583_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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,aa(real,fun(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),aa(num,num,bit0,one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3584_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xb)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).

% sin_arccos
tff(fact_3585_summable__complex__of__real,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(complex,aTP_Lamp_gu(fun(nat,real),fun(nat,complex),F2))
    <=> summable(real,F2) ) ).

% summable_complex_of_real
tff(fact_3586_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real] :
          ( ( aa(real,A,real_Vector_of_real(A),Xb) = one_one(A) )
        <=> ( Xb = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_3587_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_3588_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_3589_of__real__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,real_Vector_of_real(A),Xb)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_mult
tff(fact_3590_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),Xb)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_divide
tff(fact_3591_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),Xb)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_add
tff(fact_3592_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Nb: nat] : aa(real,A,real_Vector_of_real(A),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(real,A,real_Vector_of_real(A),Xb)),Nb) ) ).

% of_real_power
tff(fact_3593_of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(real,A,real_Vector_of_real(A),Xb)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_diff
tff(fact_3594_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3595_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_gv(fun(B,real),fun(B,A),F2)),S3) ) ).

% of_real_sum
tff(fact_3596_arccos__minus__1,axiom,
    aa(real,real,arccos,aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_3597_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_3598_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_3599_cos__arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_3600_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_3601_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: real] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),Xb)),one_one(A))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_3602_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: 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),Xb)),aa(num,A,numeral_numeral(A),B2))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(num,real,numeral_numeral(real),B2))) ) ).

% norm_of_real_addn
tff(fact_3603_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),aa(num,num,bit0,one2))) ).

% arccos_0
tff(fact_3604_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),aa(num,num,bit0,one2))) ).

% arcsin_1
tff(fact_3605_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),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3606_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),aa(num,num,bit0,one2)))) = one_one(A) ) ) ).

% sin_of_real_pi_half
tff(fact_3607_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),aa(num,num,bit0,one2)))) ).

% arcsin_minus_1
tff(fact_3608_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_3609_diffs__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F2: fun(nat,real),X4: nat] : aa(nat,A,diffs(A,aTP_Lamp_gw(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_3610_summable__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => summable(A,aTP_Lamp_gx(fun(nat,real),fun(nat,A),X5)) ) ) ).

% summable_of_real
tff(fact_3611_sums__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,real),C2: real] :
          ( aa(A,$o,sums(A,aTP_Lamp_gy(fun(nat,real),fun(nat,A),F2)),aa(real,A,real_Vector_of_real(A),C2))
        <=> aa(real,$o,sums(real,F2),C2) ) ) ).

% sums_of_real_iff
tff(fact_3612_sums__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real),A2: real] :
          ( aa(real,$o,sums(real,X5),A2)
         => aa(A,$o,sums(A,aTP_Lamp_gx(fun(nat,real),fun(nat,A),X5)),aa(real,A,real_Vector_of_real(A),A2)) ) ) ).

% sums_of_real
tff(fact_3613_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,Xb: real] :
          ( ( Y != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),Xb)),aa(real,A,real_Vector_of_real(A),Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_3614_Complex__mult__complex__of__real,axiom,
    ! [Xb: real,Y: real,R: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(Xb,Y)),aa(real,complex,real_Vector_of_real(complex),R)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),Xb),R),aa(real,real,aa(real,fun(real,real),times_times(real),Y),R)) ).

% Complex_mult_complex_of_real
tff(fact_3615_complex__of__real__mult__Complex,axiom,
    ! [R: real,Xb: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R)),complex2(Xb,Y)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),R),Y)) ).

% complex_of_real_mult_Complex
tff(fact_3616_complex__of__real__add__Complex,axiom,
    ! [R: real,Xb: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),R)),complex2(Xb,Y)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),R),Xb),Y) ).

% complex_of_real_add_Complex
tff(fact_3617_Complex__add__complex__of__real,axiom,
    ! [Xb: real,Y: real,R: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),complex2(Xb,Y)),aa(real,complex,real_Vector_of_real(complex),R)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),R),Y) ).

% Complex_add_complex_of_real
tff(fact_3618_arccos__le__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),aa(real,real,arccos,Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_3619_arccos__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Xb)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_3620_arccos__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)) )
     => ( ( aa(real,real,arccos,Xb) = aa(real,real,arccos,Y) )
      <=> ( Xb = Y ) ) ) ).

% arccos_eq_iff
tff(fact_3621_arcsin__le__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3622_arcsin__minus,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,Xb)) ) ) ) ).

% arcsin_minus
tff(fact_3623_suminf__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => ( aa(real,A,real_Vector_of_real(A),suminf(real,X5)) = suminf(A,aTP_Lamp_gx(fun(nat,real),fun(nat,A),X5)) ) ) ) ).

% suminf_of_real
tff(fact_3624_arcsin__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_3625_arcsin__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( ( aa(real,real,arcsin,Xb) = aa(real,real,arcsin,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3626_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),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,Xb))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_3627_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_3628_arccos__less__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Y)),aa(real,real,arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_3629_arccos__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Xb)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_3630_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_3631_arccos__cos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( aa(real,real,arccos,cos(real,Xb)) = Xb ) ) ) ).

% arccos_cos
tff(fact_3632_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3633_arcsin__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_3634_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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_3635_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [B2: real,A2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(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,aa(real,fun(real,real),minus_minus(real),B2),A2))) ) ).

% norm_of_real_diff
tff(fact_3636_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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_3637_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_3638_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_3639_sin__arccos__nonzero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xb)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3640_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Mb: int,Xb: real] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),ring_1_of_int(A,Mb)),aa(real,A,real_Vector_of_real(A),Xb))) = aa(real,A,real_Vector_of_real(A),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,Mb)),Xb))) ) ).

% cos_int_times_real
tff(fact_3641_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Mb: int,Xb: real] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),ring_1_of_int(A,Mb)),aa(real,A,real_Vector_of_real(A),Xb))) = aa(real,A,real_Vector_of_real(A),sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,Mb)),Xb))) ) ).

% sin_int_times_real
tff(fact_3642_arccos__cos2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Xb)
       => ( aa(real,real,arccos,cos(real,Xb)) = aa(real,real,uminus_uminus(real),Xb) ) ) ) ).

% arccos_cos2
tff(fact_3643_arccos__minus,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,Xb)) ) ) ) ).

% arccos_minus
tff(fact_3644_cos__arcsin__nonzero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xb)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3645_arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi)
          & ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_3646_arccos__minus__abs,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,Xb)) ) ) ).

% arccos_minus_abs
tff(fact_3647_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : sin(A,Xb) = cos(A,aa(A,A,aa(A,fun(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),aa(num,num,bit0,one2)))),Xb)) ) ).

% sin_cos_eq
tff(fact_3648_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : cos(A,Xb) = sin(A,aa(A,A,aa(A,fun(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),aa(num,num,bit0,one2)))),Xb)) ) ).

% cos_sin_eq
tff(fact_3649_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] : aa(A,A,arsinh(A),Xb) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))))))) ) ).

% arsinh_def
tff(fact_3650_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,uminus_uminus(A),sin(A,Xb)) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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),aa(num,num,bit0,one2))))) ) ).

% minus_sin_cos_eq
tff(fact_3651_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))) ) ) ).

% arccos_le_pi2
tff(fact_3652_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3653_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_3654_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))) ) ) ).

% arcsin_ubound
tff(fact_3655_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_3656_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,real,arcsin,sin(real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_3657_le__arcsin__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,arcsin,Xb))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3658_arcsin__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xb)),Y)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3659_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),pi)
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3660_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_3661_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa(real,real,arccos,cos(real,Theta)) != aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Theta),aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,K2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_3662_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [M5: nat,N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M5)),aa(nat,A,X5,N4)) )
            | ! [M5: nat,N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N4)),aa(nat,A,X5,M5)) ) ) ) ) ).

% monoseq_def
tff(fact_3663_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,M)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI2
tff(fact_3664_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M)),aa(nat,A,X5,N)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI1
tff(fact_3665_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) = 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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))),comm_s3205402744901411588hammer(A,Z,Nb))),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),aa(num,num,bit0,one2)))),Nb)) ) ).

% pochhammer_double
tff(fact_3666_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,one_one(nat)) = A2 ) ).

% pochhammer_1
tff(fact_3667_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_3668_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_3669_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,Nb)) ) ) ).

% pochhammer_pos
tff(fact_3670_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat,Mb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => ( comm_s3205402744901411588hammer(A,A2,Mb) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_3671_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Mb) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => ( comm_s3205402744901411588hammer(A,A2,Nb) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_3672_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = comm_s3205402744901411588hammer(A,one_one(A),Nb) ) ).

% pochhammer_fact
tff(fact_3673_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,Nb)) ) ) ).

% pochhammer_nonneg
tff(fact_3674_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] :
          comm_s3205402744901411588hammer(A,zero_zero(A),Nb) = $ite(Nb = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% pochhammer_0_left
tff(fact_3675_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = 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)),Nb)) ) ).

% pochhammer_rec
tff(fact_3676_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ).

% pochhammer_Suc
tff(fact_3677_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Nb: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb)) = 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),Nb))),comm_s3205402744901411588hammer(A,Z,Nb)) ) ).

% pochhammer_rec'
tff(fact_3678_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_3679_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Nb: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_3680_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Nb)
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_3681_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_3682_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Nb: nat,Mb: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,Nb)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Nb)),Mb)) ) ).

% pochhammer_product'
tff(fact_3683_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Mb: nat,Nb: nat,Z: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( comm_s3205402744901411588hammer(A,Z,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,Mb)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ) ) ).

% pochhammer_product
tff(fact_3684_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_3685_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),Nb) = 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))),Nb)),semiring_char_0_fact(A,Nb)) ) ).

% pochhammer_same
tff(fact_3686_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,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_3687_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,aa(A,fun(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_3688_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: fun(nat,A)] :
          ( topological_monoseq(A,A2)
         => topological_monoseq(A,aTP_Lamp_gz(fun(nat,A),fun(nat,A),A2)) ) ) ).

% monoseq_minus
tff(fact_3689_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) = 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),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb))),semiring_char_0_fact(A,Nb)) ) ).

% fact_double
tff(fact_3690_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N4)),aa(nat,A,X5,aa(nat,nat,suc,N4)))
            | ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N4))),aa(nat,A,X5,N4)) ) ) ) ).

% monoseq_Suc
tff(fact_3691_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N))),aa(nat,A,X5,N))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI2
tff(fact_3692_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,aa(nat,nat,suc,N)))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI1
tff(fact_3693_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb))),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),aa(num,num,bit0,one2)))),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ha(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),aa(num,num,bit0,one2))),Nb)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_3694_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] :
          comm_s3205402744901411588hammer(A,A2,Nb) = $ite(Nb = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_hb(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_3695_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),aa(num,num,bit0,one2))),aa(real,complex,real_Vector_of_real(complex),pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_3696_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),aa(num,num,bit0,one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_3697_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_hc(B,A)),A3) = one_one(A) ) ).

% prod.neutral_const
tff(fact_3698_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),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_hd(fun(B,nat),fun(B,A),F2)),A3) ) ).

% of_nat_prod
tff(fact_3699_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F2: fun(B,int),A3: set(B)] : ring_1_of_int(A,aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_he(fun(B,int),fun(B,A),F2)),A3) ) ).

% of_int_prod
tff(fact_3700_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),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7121269368397514597t_prod(B,real),F2),S3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_hf(fun(B,real),fun(B,A),F2)),S3) ) ).

% of_real_prod
tff(fact_3701_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_3702_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ~ finite_finite2(A,A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_3703_norm__ii,axiom,
    real_V7770717601297561774m_norm(complex,imaginary_unit) = one_one(real) ).

% norm_ii
tff(fact_3704_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_hg(A,fun(fun(A,B),fun(A,B)),A2),B2)),S) = $ite(aa(set(A),$o,member(A,A2),S),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_3705_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_hh(A,fun(fun(A,B),fun(A,B)),A2),B2)),S) = $ite(aa(set(A),$o,member(A,A2),S),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_3706_divide__i,axiom,
    ! [Xb: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,uminus_uminus(complex),imaginary_unit)),Xb) ).

% divide_i
tff(fact_3707_complex__i__mult__minus,axiom,
    ! [Xb: 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),Xb)) = aa(complex,complex,uminus_uminus(complex),Xb) ).

% complex_i_mult_minus
tff(fact_3708_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ~ aa(set(A),$o,member(A,Xb),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert(A,Xb),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ) ) ).

% prod.insert
tff(fact_3709_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% prod.lessThan_Suc
tff(fact_3710_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_3711_divide__numeral__i,axiom,
    ! [Z: complex,Nb: 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),Nb)),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),Nb)) ).

% divide_numeral_i
tff(fact_3712_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),Mb),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ).

% prod.cl_ivl_Suc
tff(fact_3713_power2__i,axiom,
    aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),imaginary_unit),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% power2_i
tff(fact_3714_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_3715_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_3716_i__even__power,axiom,
    ! [Nb: 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),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),aa(complex,complex,uminus_uminus(complex),one_one(complex))),Nb) ).

% i_even_power
tff(fact_3717_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)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(C),fun(B,A),aTP_Lamp_hi(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B4)),A3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(set(B),fun(C,A),aTP_Lamp_hk(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3)),B4) ) ).

% prod.swap
tff(fact_3718_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => ( aa(A,B,G,X2) = one_one(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_3719_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) != one_one(A) )
         => ~ ! [A4: B] :
                ( aa(set(B),$o,member(B,A4),A3)
               => ( aa(B,A,G,A4) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_3720_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F2: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7121269368397514597t_prod(B,real),aTP_Lamp_hl(fun(B,A),fun(B,real),F2)),A3)) ) ).

% norm_prod_le
tff(fact_3721_prod__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [F2: fun(A,B),A3: set(A)] : aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),aTP_Lamp_hm(fun(A,B),fun(A,real),F2)),A3) = real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ).

% prod_norm
tff(fact_3722_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_hn(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),A3)) ) ).

% prod.distrib
tff(fact_3723_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ho(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)) ) ).

% prod_dividef
tff(fact_3724_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: set(B),Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)),Nb) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(nat,fun(B,A),aTP_Lamp_hp(fun(B,A),fun(nat,fun(B,A)),F2),Nb)),A3) ) ).

% prod_power_distrib
tff(fact_3725_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),B4: set(B),G: fun(A,fun(B,C)),R2: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(B,B4)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_hq(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B4),G),R2)),A3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_hs(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R2)),B4) ) ) ) ) ).

% prod.swap_restrict
tff(fact_3726_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F2: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ht(fun(B,A),fun(B,A),F2)),A3) ) ).

% abs_prod
tff(fact_3727_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F2: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aTP_Lamp_cn(fun(B,A),fun(A,fun(B,A)),F2),A2)),A3),A2) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A3),A2) ) ).

% mod_prod_eq
tff(fact_3728_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_nonneg
tff(fact_3729_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),aa(A,B,G,I3)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ).

% prod_mono
tff(fact_3730_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_pos
tff(fact_3731_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ).

% prod_ge_1
tff(fact_3732_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F2: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F2),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_hu(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_3733_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_co(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_hv(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% prod.inter_filter
tff(fact_3734_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_3735_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)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,nat),fun(B,A),aTP_Lamp_hx(A,fun(fun(B,nat),fun(B,A)),C2),F2)),A3) ) ).

% power_sum
tff(fact_3736_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hy(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_3737_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_3738_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R2: fun(A,fun(A,$o)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,one_one(A)),one_one(A))
         => ( ! [X15: A,Y15: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R2,X15),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R2,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(A,A,aa(A,fun(A,A),times_times(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),times_times(A),X22),Y23)) )
           => ( finite_finite2(B,S)
             => ( ! [X2: B] :
                    ( aa(set(B),$o,member(B,X2),S)
                   => aa(A,$o,aa(A,fun(A,$o),R2,aa(B,A,H,X2)),aa(B,A,G,X2)) )
               => aa(A,$o,aa(A,fun(A,$o),R2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S)) ) ) ) ) ) ).

% prod.related
tff(fact_3739_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert(A,Xb),A3)) = $ite(aa(set(A),$o,member(A,Xb),A3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3))) ) ) ) ).

% prod.insert_if
tff(fact_3740_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_3741_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T3: set(B),S: set(A),I2: fun(B,A),J: fun(A,B),T2: set(B),G: fun(A,C),H: fun(B,C)] :
          ( finite_finite2(A,S4)
         => ( finite_finite2(B,T3)
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S4))
                 => ( aa(B,A,I2,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S4))
                   => aa(set(B),$o,member(B,aa(A,B,J,A4)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3)) )
               => ( ! [B3: B] :
                      ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                     => ( aa(A,B,J,aa(B,A,I2,B3)) = B3 ) )
                 => ( ! [B3: B] :
                        ( aa(set(B),$o,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                       => aa(set(A),$o,member(A,aa(B,A,I2,B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S4)) )
                   => ( ! [A4: A] :
                          ( aa(set(A),$o,member(A,A4),S4)
                         => ( aa(A,C,G,A4) = one_one(C) ) )
                     => ( ! [B3: B] :
                            ( aa(set(B),$o,member(B,B3),T3)
                           => ( aa(B,C,H,B3) = one_one(C) ) )
                       => ( ! [A4: A] :
                              ( aa(set(A),$o,member(A,A4),S)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),T2) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_3742_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_hz(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3743_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)] :
          ( finite_finite2(A,I5)
         => ( aa(B,B,exp(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),I5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aTP_Lamp_ia(fun(A,B),fun(A,B),F2)),I5) ) ) ) ).

% exp_sum
tff(fact_3744_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(nat,fun(nat,A)),G),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.nat_diff_reindex
tff(fact_3745_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ic(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% prod.atLeastAtMost_rev
tff(fact_3746_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I2: A,F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( aa(set(A),$o,member(A,I2),I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I2))
             => ( ! [I3: A] :
                    ( aa(set(A),$o,member(A,I3),I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3747_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( aa(set(A),$o,member(A,I3),I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I5)) ) ) ) ) ).

% less_1_prod
tff(fact_3748_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_3749_Complex__eq__i,axiom,
    ! [Xb: real,Y: real] :
      ( ( complex2(Xb,Y) = imaginary_unit )
    <=> ( ( Xb = zero_zero(real) )
        & ( Y = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_3750_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B4: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
         => ( finite_finite2(A,A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3751_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),S)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3752_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,H,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),S)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3753_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3754_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T2) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3755_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C5)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B4))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C5) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B4) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3756_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A3: set(A),B4: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C5)
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B4))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B4) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3757_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_3758_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3759_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3760_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_3761_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_3762_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3763_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3764_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3765_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_df(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb))) ) ).

% fact_prod
tff(fact_3766_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),A3)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I3)),aa(A,B,G,I3)) ) )
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)) ) ) ) ) ).

% prod_mono_strict
tff(fact_3767_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3))
          <=> ? [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
                & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(A,B,F2,X)) ) ) ) ) ).

% even_prod_iff
tff(fact_3768_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),insert(A,Xb),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_3769_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,Xb),A3)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_3770_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3771_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xb: fun(nat,fun(A,A)),Xaa: nat,Xba: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xb,Xaa,Xba,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xba),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xba,aa(A,A,aa(nat,fun(A,A),Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_3772_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,F2,A2,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc2,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),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_3773_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_id(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),S) = $ite(aa(set(A),$o,member(A,A2),S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_3774_complex__of__real__i,axiom,
    ! [R: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R)),imaginary_unit) = complex2(zero_zero(real),R) ).

% complex_of_real_i
tff(fact_3775_i__complex__of__real,axiom,
    ! [R: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),R)) = complex2(zero_zero(real),R) ).

% i_complex_of_real
tff(fact_3776_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z,I3))),one_one(real)) )
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),I5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W,I3))),one_one(real)) )
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Z),I5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),W),I5)))),aa(set(A),real,groups7311177749621191930dd_sum(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_ie(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_3777_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_3778_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( semiring_char_0_fact(nat,Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,Nb)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_df(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_3779_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B4: set(A),A3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,B4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
           => ( ! [B3: A] :
                  ( aa(set(A),$o,member(A,B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,B3)) )
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B4)) ) ) ) ) ) ).

% prod_mono2
tff(fact_3780_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A3: set(A),F2: fun(A,B),A2: A] :
          ( finite_finite2(A,A3)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,A2),A3),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(A,B,F2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)) ) ) ) ) ).

% prod_diff1
tff(fact_3781_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_if(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod
tff(fact_3782_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ig(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)) ) ).

% pochhammer_prod_rev
tff(fact_3783_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Mb)),semiring_char_0_fact(nat,Nb)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_df(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),Mb)) ) ) ).

% fact_div_fact
tff(fact_3784_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_3785_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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),aa(num,num,bit0,one2))),Mb),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ih(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.in_pairs
tff(fact_3786_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_ii(fun(nat,A),fun(nat,fun(A,A)),F2),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3787_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ig(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3788_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_3789_cmod__complex__polar,axiom,
    ! [R: 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),R)),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),R) ).

% cmod_complex_polar
tff(fact_3790_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb,one_one(nat))) ) ).

% fact_code
tff(fact_3791_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),aa(num,num,bit0,one2))) ).

% Arg_minus_ii
tff(fact_3792_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),aa(num,num,bit0,one2))))) ).

% csqrt_ii
tff(fact_3793_Arg__ii,axiom,
    arg(imaginary_unit) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% Arg_ii
tff(fact_3794_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),aa(num,num,bit0,one2))))) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% cis_minus_pi_half
tff(fact_3795_prod__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A3) = one_one(nat) )
      <=> ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => ( aa(A,nat,F2,X) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_3796_norm__cis,axiom,
    ! [A2: real] : real_V7770717601297561774m_norm(complex,cis(A2)) = one_one(real) ).

% norm_cis
tff(fact_3797_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A3))
      <=> ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3798_power2__csqrt,axiom,
    ! [Z: complex] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),csqrt(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Z ).

% power2_csqrt
tff(fact_3799_cis__pi__half,axiom,
    cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = imaginary_unit ).

% cis_pi_half
tff(fact_3800_cis__2pi,axiom,
    cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) = one_one(complex) ).

% cis_2pi
tff(fact_3801_int__prod,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A3)) = aa(set(A),int,aa(fun(A,int),fun(set(A),int),groups7121269368397514597t_prod(A,int),aTP_Lamp_di(fun(A,nat),fun(A,int),F2)),A3) ).

% int_prod
tff(fact_3802_cis__divide,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),cis(A2)),cis(B2)) = cis(aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)) ).

% cis_divide
tff(fact_3803_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_3804_prod__int__eq,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I2,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_br(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_3805_DeMoivre,axiom,
    ! [A2: real,Nb: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),Nb) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A2)) ).

% DeMoivre
tff(fact_3806_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_br(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)))) ).

% prod_int_plus_eq
tff(fact_3807_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( finite_finite2(A,I5)
     => ( ! [I3: A] :
            ( aa(set(A),$o,member(A,I3),I5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,I3)) )
       => ( aa(real,real,ln_ln(real),aa(set(A),real,aa(fun(A,real),fun(set(A),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_3808_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_3809_of__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,Xb)) = csqrt(aa(real,complex,real_Vector_of_real(complex),Xb)) ) ) ).

% of_real_sqrt
tff(fact_3810_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_3811_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => bij_betw(nat,complex,aTP_Lamp_ik(nat,fun(nat,complex),Nb),aa(nat,set(nat),set_ord_lessThan(nat),Nb),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_da(nat,fun(complex,$o),Nb))) ) ).

% bij_betw_roots_unity
tff(fact_3812_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa(nat,nat,binomial(Nb),K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),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,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),one_one(nat)),Nb,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_3813_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Nb) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),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),Nb),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_3814_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),Nb) = semiri8178284476397505188at_aux(A,aTP_Lamp_il(A,A),Nb,zero_zero(A)) ) ).

% of_nat_code
tff(fact_3815_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),Nb) = aa(nat,nat,suc,Nb) ).

% binomial_Suc_n
tff(fact_3816_binomial__n__n,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),Nb) = one_one(nat) ).

% binomial_n_n
tff(fact_3817_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( ring_1_of_int(A,archim6421214686448440834_floor(A,Xb)) = Xb )
        <=> ? [N4: int] : Xb = ring_1_of_int(A,N4) ) ) ).

% of_int_floor_cancel
tff(fact_3818_binomial__1,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(nat,nat,suc,zero_zero(nat))) = Nb ).

% binomial_1
tff(fact_3819_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_3820_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_3821_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_3822_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_3823_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_3824_binomial__n__0,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_3825_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_3826_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),ring_1_of_int(A,Z))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,Xb)),Z) ) ).

% floor_diff_of_int
tff(fact_3827_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) ) ).

% zero_le_floor
tff(fact_3828_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_3829_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V)),Xb) ) ) ).

% numeral_le_floor
tff(fact_3830_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ).

% zero_less_floor
tff(fact_3831_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_3832_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(num,A,numeral_numeral(A),V)) ) ) ).

% floor_less_numeral
tff(fact_3833_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ).

% one_le_floor
tff(fact_3834_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ).

% floor_less_one
tff(fact_3835_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_3836_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_diff_numeral
tff(fact_3837_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,Xb)),one_one(int)) ) ).

% floor_diff_one
tff(fact_3838_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: num,Nb: nat] : archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ).

% floor_numeral_power
tff(fact_3839_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_3840_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),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))),Xb) ) ) ).

% numeral_less_floor
tff(fact_3841_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),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_3842_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb) ) ) ).

% one_less_floor
tff(fact_3843_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ).

% floor_le_one
tff(fact_3844_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xb) ) ) ).

% neg_numeral_le_floor
tff(fact_3845_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_neg_numeral
tff(fact_3846_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_3847_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_3848_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),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))),Xb) ) ) ).

% neg_numeral_less_floor
tff(fact_3849_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),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_3850_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_3851_prod_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [H: fun(A,B),S: set(A),T2: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S,T2)
         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_im(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T2) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_3852_choose__one,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),one_one(nat)) = Nb ).

% choose_one
tff(fact_3853_sum_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [H: fun(A,B),S: set(A),T2: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S,T2)
         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_in(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T2) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_3854_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
     => ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_3855_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_3856_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,archim6421214686448440834_floor(A,Xb))),Xb) ) ).

% of_int_floor_le
tff(fact_3857_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% floor_less_cancel
tff(fact_3858_Suc__times__binomial,axiom,
    ! [K: nat,Nb: 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,Nb)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K)) ).

% Suc_times_binomial
tff(fact_3859_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).

% Suc_times_binomial_eq
tff(fact_3860_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_3861_choose__mult__lemma,axiom,
    ! [Mb: nat,R: 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),Mb),R)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),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),Mb),R)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R)),Mb)) ).

% choose_mult_lemma
tff(fact_3862_binomial__le__pow,axiom,
    ! [R: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),R)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R)) ) ).

% binomial_le_pow
tff(fact_3863_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K)) ) ).

% zero_less_binomial
tff(fact_3864_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_3865_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,Z)),Xb) ) ) ).

% le_floor_iff
tff(fact_3866_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),ring_1_of_int(A,Z)) ) ) ).

% floor_less_iff
tff(fact_3867_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),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,Nb)),aa(nat,nat,binomial(Nb),K))),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_3868_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))) ) ).

% le_floor_add
tff(fact_3869_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,Xb)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,Z)),Xb)) ) ).

% int_add_floor
tff(fact_3870_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),ring_1_of_int(A,Z))) ) ).

% floor_add_int
tff(fact_3871_choose__mult,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),Mb)),aa(nat,nat,binomial(Mb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K))) ) ) ) ).

% choose_mult
tff(fact_3872_binomial__absorb__comp,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_3873_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),ring_1_of_int(A,K)),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_3874_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Nb: nat] :
          ( ( Xb = ring_1_of_int(A,archim6421214686448440834_floor(A,Xb)) )
         => ( archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),archim6421214686448440834_floor(A,Xb)),Nb) ) ) ) ).

% floor_power
tff(fact_3875_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Nb: nat,I2: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,Nb),I2) = semiri8178284476397505188at_aux(A,Inc,Nb,aa(A,A,Inc,I2)) ) ).

% of_nat_aux.simps(2)
tff(fact_3876_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I2: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I2) = I2 ) ).

% of_nat_aux.simps(1)
tff(fact_3877_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) ) ).

% one_add_floor
tff(fact_3878_binomial__absorption,axiom,
    ! [K: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_3879_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Mb: nat,Nb: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)) ) ).

% floor_divide_of_nat_eq
tff(fact_3880_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_ceiling(A,Xb) = $ite(Xb = ring_1_of_int(A,archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Xb),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),one_one(int))) ) ).

% ceiling_altdef
tff(fact_3881_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T3: set(B),H: fun(A,B),S: set(A),T2: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S4)
         => ( finite_finite2(B,T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),S4)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = zero_zero(C) ) )
               => ( ! [B3: B] :
                      ( aa(set(B),$o,member(B,B3),T3)
                     => ( aa(B,C,G,B3) = zero_zero(C) ) )
                 => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_in(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T2) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_3882_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xb)),archim6421214686448440834_floor(A,Xb))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_3883_real__of__int__floor__add__one__gt,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),R),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,archim6421214686448440834_floor(real,R))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_3884_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),ring_1_of_int(real,Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,Nb)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_3885_real__of__int__floor__add__one__ge,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,archim6421214686448440834_floor(real,R))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_3886_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T3: set(B),H: fun(A,B),S: set(A),T2: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S4)
         => ( finite_finite2(B,T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),S4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
             => ( ! [A4: A] :
                    ( aa(set(A),$o,member(A,A4),S4)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = one_one(C) ) )
               => ( ! [B3: B] :
                      ( aa(set(B),$o,member(B,B3),T3)
                     => ( aa(B,C,G,B3) = one_one(C) ) )
                 => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_im(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T2) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_3887_real__of__int__floor__gt__diff__one,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R),one_one(real))),ring_1_of_int(real,archim6421214686448440834_floor(real,R))) ).

% real_of_int_floor_gt_diff_one
tff(fact_3888_real__of__int__floor__ge__diff__one,axiom,
    ! [R: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R),one_one(real))),ring_1_of_int(real,archim6421214686448440834_floor(real,R))) ).

% real_of_int_floor_ge_diff_one
tff(fact_3889_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( 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,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))),aa(nat,nat,binomial(Nb),K)) = semiring_char_0_fact(nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_3890_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,Z)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,Z)),one_one(A)))
           => ( archim6421214686448440834_floor(A,Xb) = Z ) ) ) ) ).

% floor_unique
tff(fact_3891_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,Xb) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,A2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,A2)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_3892_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,I4)),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,I4)),one_one(A))) )
             => aa(int,$o,P,I4) ) ) ) ).

% floor_split
tff(fact_3893_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$o,aa(A,fun(A,$o),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),Nb)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_3894_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),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_3895_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,Z)),one_one(A))),Xb) ) ) ).

% less_floor_iff
tff(fact_3896_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,Z)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_3897_binomial__mono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_mono
tff(fact_3898_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),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),aa(num,num,bit0,one2))),Nb)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_3899_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% binomial_maximum
tff(fact_3900_binomial__antimono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),K)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_3901_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% binomial_le_pow2
tff(fact_3902_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ring_1_of_int(A,archim6421214686448440834_floor(A,Xb))),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_3903_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_3904_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_3905_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ring_1_of_int(real,Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,Nb)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_3906_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Nb)),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,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_3907_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),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_3908_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(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_3909_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_3910_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K7)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_strict_mono
tff(fact_3911_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,binomial(Nb),K7)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_3912_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( aa(nat,nat,binomial(Nb),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% central_binomial_odd
tff(fact_3913_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_3914_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Nb)),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,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)))) ) ) ) ).

% binomial_fact
tff(fact_3915_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( 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(Nb),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))) ) ) ) ).

% fact_binomial
tff(fact_3916_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$o),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),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_3917_choose__two,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% choose_two
tff(fact_3918_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_round(A,Xb) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% round_def
tff(fact_3919_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),Xb)) = K )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,ring_1_of_int(real,K))),Xb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,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_3920_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_3921_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_3922_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,aa(num,num,bit0,one2)))),Nb)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb)))),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),aa(num,num,bit0,one2))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_3923_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ finite_finite2(A,A3)
     => ? [H3: fun(A,A)] : bij_betw(A,A,H3,A3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_3924_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_io(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) ) ) ) ).

% choose_odd_sum
tff(fact_3925_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ip(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) ) ) ) ).

% choose_even_sum
tff(fact_3926_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_round(A,Xb) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,Xb)),archimedean_ceiling(A,Xb),archim6421214686448440834_floor(A,Xb)) ) ).

% round_altdef
tff(fact_3927_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( aa(set(A),$o,member(A,I2),aa(A,set(A),set_ord_atMost(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),K) ) ) ).

% atMost_iff
tff(fact_3928_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),Xb)),aa(A,set(A),set_ord_atMost(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% atMost_subset_iff
tff(fact_3929_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atMost(A),H2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),H),H2) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_3930_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% sum.atMost_Suc
tff(fact_3931_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% prod.atMost_Suc
tff(fact_3932_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_iq(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_3933_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_atMost(nat),K) ).

% lessThan_Suc_atMost
tff(fact_3934_atMost__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% atMost_Suc
tff(fact_3935_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,Xb)) ) ).

% frac_ge_0
tff(fact_3936_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,Xb)),one_one(A)) ) ).

% frac_lt_1
tff(fact_3937_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) = archimedean_frac(A,Xb) ) ).

% frac_1_eq
tff(fact_3938_atMost__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_atMost(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_3939_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A2)),aa(A,set(A),set_ord_lessThan(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3940_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ir(nat,fun(nat,nat),Mb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,Mb)) ).

% sum_choose_upper
tff(fact_3941_frac__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_frac(A,Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),ring_1_of_int(A,archim6421214686448440834_floor(A,Xb))) ) ).

% frac_def
tff(fact_3942_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = 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_cv(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% sum.atMost_Suc_shift
tff(fact_3943_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),I2: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_du(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_atMost(nat),I2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,aa(nat,nat,suc,I2))) ) ).

% sum_telescope
tff(fact_3944_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,D2: fun(nat,A)] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_is(fun(nat,A),fun(A,fun(nat,A)),C2),X)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_is(fun(nat,A),fun(A,fun(nat,A)),D2),X)),aa(nat,set(nat),set_ord_atMost(nat),Nb))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
             => ( aa(nat,A,C2,I4) = aa(nat,A,D2,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_3945_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B4: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),aa(nat,set(nat),set_ord_atMost(nat),N))),B4)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_3946_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% prod.atMost_Suc_shift
tff(fact_3947_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_it(fun(nat,fun(nat,A)),fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iv(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.nested_swap'
tff(fact_3948_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_iw(fun(nat,fun(nat,A)),fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_iy(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.nested_swap'
tff(fact_3949_sum__choose__lower,axiom,
    ! [R: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_iz(nat,fun(nat,nat),R)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R),Nb))),Nb) ).

% sum_choose_lower
tff(fact_3950_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ja(nat,fun(nat,nat),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = 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),Nb),Mb)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_3951_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ja(nat,fun(nat,nat),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = 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),Nb),Mb)),one_one(nat))),Mb) ).

% choose_rising_sum(2)
tff(fact_3952_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( archimedean_frac(A,Xb) = Xb )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_3953_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_is(fun(nat,A),fun(A,fun(nat,A)),C2),X)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
             => ( aa(nat,A,C2,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_3954_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jb(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_3955_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_3956_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = 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_cv(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% sum.atMost_shift
tff(fact_3957_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),aa(nat,set(nat),set_ord_atMost(nat),Mb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)))) ) ).

% sum_up_index_split
tff(fact_3958_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atMost(nat),Nb)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_3959_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ).

% prod.atMost_shift
tff(fact_3960_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_jc(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_je(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% sum.triangle_reindex_eq
tff(fact_3961_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_jc(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jg(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% prod.triangle_reindex_eq
tff(fact_3962_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_jh(nat,fun(nat,fun(nat,nat)),Mb),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_3963_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ji(nat,fun(nat,fun(nat,fun(nat,nat))),Mb),Nb),R)),aa(nat,set(nat),set_ord_atMost(nat),R)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),R) ).

% vandermonde
tff(fact_3964_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb))) ) ).

% sum_gp_basic
tff(fact_3965_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
        <=> ? [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Nb)
              & ( aa(nat,A,C2,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_3966_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb))) ) ) ) ).

% polyfun_roots_finite
tff(fact_3967_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: A,Nb: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) )
         => ~ ! [B3: fun(nat,A)] :
                ~ ! [Z2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),B3),Z2)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ).

% polyfun_linear_factor_root
tff(fact_3968_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Nb: nat,A2: A] :
        ? [B3: fun(nat,A)] :
        ! [Z2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = 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),minus_minus(A),Z2),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),B3),Z2)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ) ).

% polyfun_linear_factor
tff(fact_3969_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Mb: nat,Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))) ) ) ) ).

% sum_power_shift
tff(fact_3970_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_jl(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_je(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% sum.triangle_reindex
tff(fact_3971_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_jl(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_jg(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) ) ).

% prod.triangle_reindex
tff(fact_3972_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_jm(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_jm(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_3973_choose__row__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) ).

% choose_row_sum
tff(fact_3974_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_jm(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_jm(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_jo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_3975_binomial,axiom,
    ! [A2: nat,B2: nat,Nb: 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)),Nb) = 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))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ).

% binomial
tff(fact_3976_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_12)
     => ( ! [X2: A,N: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N),X2)
           => ? [Y5: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N)),Y5)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N),X2),Y5) ) )
       => ? [F3: fun(nat,A)] :
          ! [N8: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N8),aa(nat,A,F3,N8))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N8),aa(nat,A,F3,N8)),aa(nat,A,F3,aa(nat,nat,suc,N8))) ) ) ) ).

% dependent_nat_choice
tff(fact_3977_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),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),aa(num,num,bit0,one2))),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_de(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% sum.in_pairs_0
tff(fact_3978_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Mb: nat,A2: fun(nat,A),Nb: nat,B2: fun(nat,A),Xb: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),I3)
             => ( aa(nat,A,A2,I3) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),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_jk(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Mb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),B2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_jr(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ) ) ) ).

% polynomial_product
tff(fact_3979_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),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),aa(num,num,bit0,one2))),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ih(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% prod.in_pairs_0
tff(fact_3980_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: A] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_is(fun(nat,A),fun(A,fun(nat,A)),C2),X)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X: nat] :
                ( aa(set(nat),$o,member(nat,X),set_or1337092689740270186AtMost(nat,one_one(nat),Nb))
               => ( aa(nat,A,C2,X) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_3981_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,Nb: 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)),Nb) = 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),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% binomial_ring
tff(fact_3982_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,B2: A,Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),Nb) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_jt(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ).

% pochhammer_binomial_sum
tff(fact_3983_polynomial__product__nat,axiom,
    ! [Mb: nat,A2: fun(nat,nat),Nb: nat,B2: fun(nat,nat),Xb: nat] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),I3)
         => ( aa(nat,nat,A2,I3) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),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_ju(fun(nat,nat),fun(nat,fun(nat,nat)),A2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Mb))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ju(fun(nat,nat),fun(nat,fun(nat,nat)),B2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_jw(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_3984_choose__square__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_jx(nat,fun(nat,nat),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),Nb) ).

% choose_square_sum
tff(fact_3985_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_jm(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_jm(fun(nat,A),fun(nat,real),B2))
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jo(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_3986_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,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,Xb)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_3987_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),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_jy(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),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_jz(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_3988_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P2)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ka(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_kb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_3989_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,Z: A,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb) = A2 )
          <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_kc(nat,fun(A,fun(A,fun(nat,A))),Nb),Z),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_3990_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,Nb)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xb))) ) ).

% sum_gp0
tff(fact_3991_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( ( Nb != one_one(nat) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kd(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3992_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),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_kf(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ) ).

% polyfun_diff_alt
tff(fact_3993_binomial__r__part__sum,axiom,
    ! [Mb: 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),aa(num,num,bit0,one2))),Mb)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb)) ).

% binomial_r_part_sum
tff(fact_3994_choose__linear__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_kg(nat,fun(nat,nat),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ).

% choose_linear_sum
tff(fact_3995_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kh(nat,fun(nat,A),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3996_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E: real,C2: fun(nat,A),Nb: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => ? [M7: real] :
            ! [Z2: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M7),real_V7770717601297561774m_norm(A,Z2))
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)))),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,Z2)),aa(nat,nat,suc,Nb)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_3997_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),aa(nat,set(nat),set_ord_atMost(nat),Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),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_kj(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ) ) ) ).

% polyfun_diff
tff(fact_3998_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_kl(A,fun(A,fun(nat,A)),Xb),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_3999_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_kn(A,fun(A,fun(nat,A)),Xb),Y)),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_4000_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_kp(A,fun(A,fun(nat,A)),Xb),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_4001_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kq(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = 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),Mb)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_4002_of__nat__id,axiom,
    ! [Nb: nat] : aa(nat,nat,semiring_1_of_nat(nat),Nb) = Nb ).

% of_nat_id
tff(fact_4003_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [A2: real,Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),Y) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) ) ).

% mult_scaleR_left
tff(fact_4004_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [Xb: A,A2: real,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Xb),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),Xb),Y)) ) ).

% mult_scaleR_right
tff(fact_4005_scaleR__one,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,one_one(real)),Xb) = Xb ) ).

% scaleR_one
tff(fact_4006_scaleR__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)),Xb) ) ).

% scaleR_scaleR
tff(fact_4007_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_4008_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_4009_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: real,Y: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,real_V8093663219630862766scaleR(A,Xb),Y)),Nb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) ) ).

% scaleR_power
tff(fact_4010_gbinomial__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [K: nat] : aa(nat,A,gbinomial(A,zero_zero(A)),aa(nat,nat,suc,K)) = zero_zero(A) ) ).

% gbinomial_0(2)
tff(fact_4011_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_4012_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_4013_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),one_one(real))),Xb) = aa(A,A,uminus_uminus(A),Xb) ) ).

% scaleR_minus1_left
tff(fact_4014_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,aa(real,fun(real,real),minus_minus(real),one_one(real)),U)),A2)),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = A2 ) ).

% scaleR_collapse
tff(fact_4015_norm__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: real,Xb: A] : real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),A2)),real_V7770717601297561774m_norm(A,Xb)) ) ).

% norm_scaleR
tff(fact_4016_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_4017_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_4018_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_4019_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),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)) = A2 ) ).

% scaleR_half_double
tff(fact_4020_scaleR__right__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ).

% scaleR_right_diff_distrib
tff(fact_4021_real__scaleR__def,axiom,
    ! [A2: real,Xb: real] : aa(real,real,real_V8093663219630862766scaleR(real,A2),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),Xb) ).

% real_scaleR_def
tff(fact_4022_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ).

% scaleR_right_distrib
tff(fact_4023_scaleR__sum__right,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,F2: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),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_kr(real,fun(fun(B,A),fun(B,A)),A2),F2)),A3) ) ).

% scaleR_sum_right
tff(fact_4024_scaleR__right_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,G: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kr(real,fun(fun(B,A),fun(B,A)),A2),G)),A3) ) ).

% scaleR_right.sum
tff(fact_4025_summable__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R: real] :
          ( summable(A,X5)
         => summable(A,aa(real,fun(nat,A),aTP_Lamp_ks(fun(nat,A),fun(real,fun(nat,A)),X5),R)) ) ) ).

% summable_scaleR_right
tff(fact_4026_sums__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),A2: A,R: real] :
          ( aa(A,$o,sums(A,X5),A2)
         => aa(A,$o,sums(A,aa(real,fun(nat,A),aTP_Lamp_ks(fun(nat,A),fun(real,fun(nat,A)),X5),R)),aa(A,A,real_V8093663219630862766scaleR(A,R),A2)) ) ) ).

% sums_scaleR_right
tff(fact_4027_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),Xaa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Xb),Xaa)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xaa)) ) ).

% scaleR_left.add
tff(fact_4028_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ).

% scaleR_left_distrib
tff(fact_4029_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,R),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(real,A,real_Vector_of_real(A),R)),Xb) ) ).

% scaleR_conv_of_real
tff(fact_4030_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R: real] : aa(real,A,real_Vector_of_real(A),R) = aa(A,A,real_V8093663219630862766scaleR(A,R),one_one(A)) ) ).

% of_real_def
tff(fact_4031_scaleR__left__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ).

% scaleR_left_diff_distrib
tff(fact_4032_scaleR__left_Odiff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y)),Xaa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,Xb),Xaa)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xaa)) ) ).

% scaleR_left.diff
tff(fact_4033_complex__scaleR,axiom,
    ! [R: real,A2: real,B2: real] : aa(complex,complex,real_V8093663219630862766scaleR(complex,R),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R),A2),aa(real,real,aa(real,fun(real,real),times_times(real),R),B2)) ).

% complex_scaleR
tff(fact_4034_scaleR__sum__left,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [F2: fun(B,real),A3: set(B),Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,F2),A3)),Xb) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_kt(fun(B,real),fun(A,fun(B,A)),F2),Xb)),A3) ) ).

% scaleR_sum_left
tff(fact_4035_scaleR__left_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [G: fun(B,real),A3: set(B),Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,G),A3)),Xb) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_kt(fun(B,real),fun(A,fun(B,A)),G),Xb)),A3) ) ).

% scaleR_left.sum
tff(fact_4036_suminf__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R: real] :
          ( summable(A,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,R),suminf(A,X5)) = suminf(A,aa(real,fun(nat,A),aTP_Lamp_ks(fun(nat,A),fun(real,fun(nat,A)),X5),R)) ) ) ) ).

% suminf_scaleR_right
tff(fact_4037_summable__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),Xb: A] :
          ( summable(real,X5)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ku(fun(nat,real),fun(A,fun(nat,A)),X5),Xb)) ) ) ).

% summable_scaleR_left
tff(fact_4038_sums__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),A2: real,Xb: A] :
          ( aa(real,$o,sums(real,X5),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_ku(fun(nat,real),fun(A,fun(nat,A)),X5),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)) ) ) ).

% sums_scaleR_left
tff(fact_4039_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_4040_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ) ) ).

% scaleR_right_mono
tff(fact_4041_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_4042_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_4043_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_4044_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),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_4045_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_4046_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_4047_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),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,aa(real,fun(real,real),minus_minus(real),B2),A2)),E)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_4048_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),E)),C2)),D2) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_4049_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_4050_suminf__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),Xb: A] :
          ( summable(real,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,suminf(real,X5)),Xb) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_ku(fun(nat,real),fun(A,fun(nat,A)),X5),Xb)) ) ) ) ).

% suminf_scaleR_left
tff(fact_4051_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_4052_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_4053_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_4054_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_4055_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_4056_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_4057_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ).

% split_scaleR_pos_le
tff(fact_4058_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_4059_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D2)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_4060_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_4061_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),one_one(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),Xb) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_4062_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb) ) ).

% scaleR_2
tff(fact_4063_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,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_4064_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,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_4065_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_4066_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_4067_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A2)
         => aa(A,$o,aa(A,fun(A,$o),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_4068_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_kv(A,fun(nat,A),Xb)),sin(A,Xb)) ) ).

% sin_converges
tff(fact_4069_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : sin(A,X4) = suminf(A,aTP_Lamp_kv(A,fun(nat,A),X4)) ) ).

% sin_def
tff(fact_4070_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_kw(A,fun(nat,A),Xb)),cos(A,Xb)) ) ).

% cos_converges
tff(fact_4071_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X4: A] : cos(A,X4) = suminf(A,aTP_Lamp_kw(A,fun(nat,A),X4)) ) ).

% cos_def
tff(fact_4072_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_kx(A,fun(nat,real),Xb)) ) ).

% summable_norm_sin
tff(fact_4073_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_ky(A,fun(nat,real),Xb)) ) ).

% summable_norm_cos
tff(fact_4074_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_4075_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,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_4076_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Mb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Mb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),Mb)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Mb)),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,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_4077_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kz(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = 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),Nb))),one_one(A))),Nb) ) ).

% gbinomial_parallel_sum
tff(fact_4078_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_la(A,fun(nat,A),Xb)),sin(A,Xb)) ) ).

% sin_minus_converges
tff(fact_4079_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_lb(A,fun(nat,A),Xb)),cos(A,Xb)) ) ).

% cos_minus_converges
tff(fact_4080_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_4081_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_4082_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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),K)),A2)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_4083_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: 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,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb))),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))),Nb)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),Nb)) ) ).

% gbinomial_index_swap
tff(fact_4084_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,aa(A,fun(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_4085_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),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,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_4086_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_4087_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,aa(A,fun(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_4088_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lc(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = 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))),Mb)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),Mb)) ) ).

% gbinomial_sum_lower_neg
tff(fact_4089_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ld(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_4090_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat,A2: A,Xb: 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_le(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = 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_lf(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4091_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),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_lg(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_4092_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),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,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_4093_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lh(nat,fun(nat,A),Mb)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb) ) ).

% gbinomial_sum_nat_pow2
tff(fact_4094_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat,A2: A,Xb: 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_le(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = 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_li(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),aa(nat,set(nat),set_ord_atMost(nat),Mb)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_4095_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] :
          aa(nat,A,gbinomial(A,A2),K) = $ite(K = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_lj(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K))) ) ).

% gbinomial_code
tff(fact_4096_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R: A,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_kq(A,fun(nat,A),R)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Mb)) = 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,Mb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,R),aa(nat,nat,suc,Mb))) ) ).

% gchoose_row_sum_weighted
tff(fact_4097_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: 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),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),Mb))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),Mb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb)) ) ).

% gbinomial_r_part_sum
tff(fact_4098_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => 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(Nb),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),Nb))))),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_da(nat,fun(complex,$o),Nb)),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_lk(complex,fun(nat,fun(complex,$o)),C2),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_4099_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,exp(A),Xb) = 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)),Xb)),suminf(A,aTP_Lamp_ll(A,fun(nat,A),Xb))) ) ).

% exp_first_two_terms
tff(fact_4100_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),sin(real,Xb)),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fz(real,fun(nat,real),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Xb)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_4101_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2)))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cot(real),Xb)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_4102_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_4103_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_4104_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A] :
          ( ( aa(A,A,inverse_inverse(A),Xb) = one_one(A) )
        <=> ( Xb = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_4105_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_4106_real__root__zero,axiom,
    ! [Nb: nat] : aa(real,real,root(Nb),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_4107_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_4108_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_4109_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% inverse_positive_iff_positive
tff(fact_4110_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_4111_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_4112_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_4113_real__root__Suc__0,axiom,
    ! [Xb: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),Xb) = Xb ).

% real_root_Suc_0
tff(fact_4114_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = aa(real,real,root(Nb),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_4115_root__0,axiom,
    ! [Xb: real] : aa(real,real,root(zero_zero(nat)),Xb) = zero_zero(real) ).

% root_0
tff(fact_4116_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_4117_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_4118_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_4119_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_4120_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_4121_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = zero_zero(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_4122_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_4123_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_4124_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_4125_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = one_one(real) )
      <=> ( Xb = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_4126_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_4127_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_4128_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_4129_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_4130_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_4131_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_4132_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_4133_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_4134_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_4135_cot__npi,axiom,
    ! [Nb: 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),Nb)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_4136_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_4137_cot__periodic,axiom,
    ! [Xb: real] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = aa(real,real,cot(real),Xb) ).

% cot_periodic
tff(fact_4138_real__root__inverse,axiom,
    ! [Nb: nat,Xb: real] : aa(real,real,root(Nb),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,root(Nb),Xb)) ).

% real_root_inverse
tff(fact_4139_real__root__commute,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa(real,real,root(Mb),aa(real,real,root(Nb),Xb)) = aa(real,real,root(Nb),aa(real,real,root(Mb),Xb)) ).

% real_root_commute
tff(fact_4140_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),A2)),Nb) = aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_inverse
tff(fact_4141_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,Xb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_4142_real__root__mult__exp,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa(real,real,root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),Xb) = aa(real,real,root(Mb),aa(real,real,root(Nb),Xb)) ).

% real_root_mult_exp
tff(fact_4143_real__root__mult,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa(real,real,root(Nb),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ).

% real_root_mult
tff(fact_4144_real__sqrt__inverse,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)) ).

% real_sqrt_inverse
tff(fact_4145_real__root__minus,axiom,
    ! [Nb: nat,Xb: real] : aa(real,real,root(Nb),aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,root(Nb),Xb)) ).

% real_root_minus
tff(fact_4146_real__root__divide,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa(real,real,root(Nb),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ).

% real_root_divide
tff(fact_4147_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R),real_V7770717601297561774m_norm(A,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xb))),aa(real,real,inverse_inverse(real),R)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_4148_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less
tff(fact_4149_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less
tff(fact_4150_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_4151_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_4152_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_4153_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_4154_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_4155_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ).

% positive_imp_inverse_positive
tff(fact_4156_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_4157_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_4158_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_4159_real__root__pos__pos__le,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_4160_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_4161_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_4162_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_4163_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_4164_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(A,A,inverse_inverse(A),Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)) ) ).

% power_mult_inverse_distrib
tff(fact_4165_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),Nb)) = 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),Xb)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb)) ) ).

% power_mult_power_inverse_commute
tff(fact_4166_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: nat,Xb: 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),Xaa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xaa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_4167_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: int,Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),ring_1_of_int(A,Xaa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),ring_1_of_int(A,Xaa))) ) ).

% mult_inverse_of_int_commute
tff(fact_4168_divide__real__def,axiom,
    ! [Xb: real,Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y) = aa(real,real,aa(real,fun(real,real),times_times(real),Xb),aa(real,real,inverse_inverse(real),Y)) ).

% divide_real_def
tff(fact_4169_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X4: nat] : aa(nat,A,diffs(A,aTP_Lamp_lm(nat,A)),X4) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X4)) ) ).

% exp_fdiffs
tff(fact_4170_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le
tff(fact_4171_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le
tff(fact_4172_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_4173_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),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_4174_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Xb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ) ).

% inverse_le_1_iff
tff(fact_4175_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_less_inverse
tff(fact_4176_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),Xb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_4177_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_4178_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_4179_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_4180_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_4181_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,aa(A,fun(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,aa(A,fun(A,A),minus_minus(A),B2),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_4182_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_4183_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_4184_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_4185_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(real,real,abs_abs(real),Xb)) = aa(real,real,abs_abs(real),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_4186_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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_4187_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_4188_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_4189_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_le_inverse
tff(fact_4190_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),Xb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ) ).

% inverse_less_1_iff
tff(fact_4191_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),Xb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_4192_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_4193_inverse__diff__inverse,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),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,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_4194_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(N2),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4195_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),Xb) ) ) ).

% reals_Archimedean
tff(fact_4196_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% sqrt_def
tff(fact_4197_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,abs_abs(real),aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Nb))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_4198_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Y: A,Xb: A,C2: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( Y = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Mb),Xb)),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),C2)) = Xb ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_4199_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Xb: A,C2: A,Y: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Mb),Xb)),C2) = Y )
          <=> ( Xb = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb)),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_4200_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_4201_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_4202_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_4203_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_4204_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_4205_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_4206_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_4207_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_4208_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D3),E2)
         => ( aa(real,$o,P,D3)
           => aa(real,$o,P,E2) ) )
     => ( ! [N: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => aa(real,$o,P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_4209_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D3),E2)
         => ( aa(real,$o,P,D3)
           => aa(real,$o,P,E2) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero(nat) )
           => aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N))) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => aa(real,$o,P,E) ) ) ) ).

% forall_pos_mono
tff(fact_4210_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero(nat) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N4)))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N4))),E) ) ) ).

% real_arch_inverse
tff(fact_4211_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,Xb)),Xb) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_4212_ln__inverse,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_inverse
tff(fact_4213_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_ln(A,fun(nat,A),Xb)) ) ).

% summable_exp
tff(fact_4214_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_lo(A,fun(nat,A),Xb)) ) ).

% summable_exp_generic
tff(fact_4215_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_4216_real__root__strict__increasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(N2),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4217_real__root__decreasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N2),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_4218_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_4219_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Nb) = Xb )
         => ( aa(real,real,root(Nb),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4220_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_4221_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = Xb ) ) ).

% odd_real_root_power_cancel
tff(fact_4222_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Nb) = Xb )
       => ( aa(real,real,root(Nb),Xb) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_4223_odd__real__root__pow,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),Nb) = Xb ) ) ).

% odd_real_root_pow
tff(fact_4224_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => ? [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N))),Xb) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_4225_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),Mb)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_4226_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_4227_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),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_4228_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),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_4229_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),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_4230_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),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_4231_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),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_4232_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),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_4233_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),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_4234_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),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_4235_log__inverse,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_4236_real__root__increasing,axiom,
    ! [Nb: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(N2),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_4237_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_lo(A,fun(nat,A),Xb)),aa(A,A,exp(A),Xb)) ) ).

% exp_converges
tff(fact_4238_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_lo(A,fun(nat,A),X4)) ) ).

% exp_def
tff(fact_4239_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_lp(A,fun(nat,real),Xb)) ) ).

% summable_norm_exp
tff(fact_4240_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,exp(real),Xb)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xb)))) ).

% exp_plus_inverse_exp
tff(fact_4241_log__root,axiom,
    ! [Nb: nat,A2: real,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( aa(real,real,log(B2),aa(real,real,root(Nb),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),Nb)) ) ) ) ).

% log_root
tff(fact_4242_log__base__root,axiom,
    ! [Nb: nat,B2: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,log(aa(real,real,root(Nb),B2)),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,log(B2),Xb)) ) ) ) ).

% log_base_root
tff(fact_4243_ln__root,axiom,
    ! [Nb: nat,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Nb),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),Nb)) ) ) ) ).

% ln_root
tff(fact_4244_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,inverse_inverse(real),Xb))) ) ).

% plus_inverse_ge_2
tff(fact_4245_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(real,real,inverse_inverse(real),Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_4246_tan__cot,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),Xb)) ).

% tan_cot
tff(fact_4247_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => ( aa(real,real,root(Nb),Xb) = powr(real,Xb,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_4248_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,exp(real),Xb)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xb)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ).

% real_le_x_sinh
tff(fact_4249_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,exp(real),Xb)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xb)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% real_le_abs_sinh
tff(fact_4250_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_lq(A,fun(A,fun(nat,fun(nat,A))),Xb),Y),Nb)),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4251_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,exp(A),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_lr(A,fun(nat,A),Xb))) ) ).

% exp_first_term
tff(fact_4252_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != 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),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),cos(A,Xb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ) ).

% tan_sec
tff(fact_4253_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cot(real),Xb)) ) ) ).

% cot_gt_zero
tff(fact_4254_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,K: nat] : aa(A,A,exp(A),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_lo(A,fun(nat,A),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ls(A,fun(nat,fun(nat,A)),Xb),K))) ) ).

% exp_first_terms
tff(fact_4255_tan__cot_H,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Xb)) = aa(real,real,cot(real),Xb) ).

% tan_cot'
tff(fact_4256_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_lt(A,fun(nat,A),Xb)),sinh(A,Xb)) ) ).

% sinh_converges
tff(fact_4257_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_lu(A,fun(nat,A),Xb)),cosh(A,Xb)) ) ).

% cosh_converges
tff(fact_4258_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_lv(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_4259_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_lw(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_4260_sinh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ).

% sinh_real_less_iff
tff(fact_4261_sinh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xb)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% sinh_real_le_iff
tff(fact_4262_sinh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_4263_sinh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sinh(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb) ) ).

% sinh_real_pos_iff
tff(fact_4264_sinh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sinh(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% sinh_real_nonneg_iff
tff(fact_4265_sinh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_4266_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_4267_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tanh(A),Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sinh(A,Xb)),cosh(A,Xb)) ) ).

% tanh_def
tff(fact_4268_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cosh(A,Xb)),sinh(A,Xb)) = aa(A,A,exp(A),Xb) ) ).

% cosh_plus_sinh
tff(fact_4269_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sinh(A,Xb)),cosh(A,Xb)) = aa(A,A,exp(A),Xb) ) ).

% sinh_plus_cosh
tff(fact_4270_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),sinh(A,Y))) ) ).

% sinh_add
tff(fact_4271_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),sinh(A,Y))) ) ).

% cosh_add
tff(fact_4272_sinh__le__cosh__real,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xb)),cosh(real,Xb)) ).

% sinh_le_cosh_real
tff(fact_4273_sinh__less__cosh__real,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xb)),cosh(real,Xb)) ).

% sinh_less_cosh_real
tff(fact_4274_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),sinh(A,Y))) ) ).

% sinh_diff
tff(fact_4275_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),sinh(A,Y))) ) ).

% cosh_diff
tff(fact_4276_sinh__minus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),sinh(A,Xb)),cosh(A,Xb)) = aa(A,A,uminus_uminus(A),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb))) ) ).

% sinh_minus_cosh
tff(fact_4277_cosh__minus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),cosh(A,Xb)),sinh(A,Xb)) = aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb)) ) ).

% cosh_minus_sinh
tff(fact_4278_cosh__real__pos,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cosh(real,Xb)) ).

% cosh_real_pos
tff(fact_4279_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_4280_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_4281_cosh__real__nonneg,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cosh(real,Xb)) ).

% cosh_real_nonneg
tff(fact_4282_cosh__real__ge__1,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),cosh(real,Xb)) ).

% cosh_real_ge_1
tff(fact_4283_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : sinh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb)) = 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),aa(num,num,bit0,one2))),sinh(A,Xb))),cosh(A,Xb)) ) ).

% sinh_double
tff(fact_4284_divide__complex__def,axiom,
    ! [Xb: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),Y) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xb),aa(complex,complex,inverse_inverse(complex),Y)) ).

% divide_complex_def
tff(fact_4285_ln__real__def,axiom,
    ! [Xb: real] : aa(real,real,ln_ln(real),Xb) = the(real,aTP_Lamp_lx(real,fun(real,$o),Xb)) ).

% ln_real_def
tff(fact_4286_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_4287_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_4288_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_4289_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_4290_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,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,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_4291_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_4292_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_4293_arcosh__cosh__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,real,arcosh(real),cosh(real,Xb)) = Xb ) ) ).

% arcosh_cosh_real
tff(fact_4294_ln__neg__is__const,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,real,ln_ln(real),Xb) = the(real,aTP_Lamp_ly(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_4295_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cosh_double
tff(fact_4296_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_lz(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_4297_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cosh(A,Xb) != 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),Xb),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),Xb)),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),Xb)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_4298_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sinh(A,Xb) = zero_zero(A) )
        <=> aa(set(A),$o,member(A,aa(A,A,exp(A),Xb)),aa(set(A),set(A),insert(A,one_one(A)),aa(set(A),set(A),insert(A,aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_4299_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),aa(num,num,bit0,one2))) ) ).

% cosh_field_def
tff(fact_4300_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),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% complex_inverse
tff(fact_4301_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,aa(A,fun(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),aa(num,num,bit0,one2))) ) ).

% sinh_field_def
tff(fact_4302_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cosh(A,Xb) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_4303_pi__half,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) = the(real,aTP_Lamp_ma(real,$o)) ).

% pi_half
tff(fact_4304_pi__def,axiom,
    pi = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),the(real,aTP_Lamp_ma(real,$o))) ).

% pi_def
tff(fact_4305_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : cosh(A,Xb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb)))) ) ).

% cosh_def
tff(fact_4306_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( cosh(real,aa(real,real,ln_ln(real),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,inverse_inverse(real),Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% cosh_ln_real
tff(fact_4307_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sinh(A,Xb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb)))) ) ).

% sinh_def
tff(fact_4308_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( sinh(real,aa(real,real,ln_ln(real),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),aa(real,real,inverse_inverse(real),Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% sinh_ln_real
tff(fact_4309_old_Orec__prod__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,X4: fun(A,fun(B,C)),Xa: product_prod(A,B)] : product_rec_prod(A,B,C,X4,Xa) = the(C,product_rec_set_prod(A,B,C,X4,Xa)) ).

% old.rec_prod_def
tff(fact_4310_the__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A2 ) )
       => ( the(A,P) = A2 ) ) ) ).

% the_equality
tff(fact_4311_the__eq__trivial,axiom,
    ! [A: $tType,A2: A] : the(A,aTP_Lamp_ek(A,fun(A,$o),A2)) = A2 ).

% the_eq_trivial
tff(fact_4312_the__sym__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : the(A,fequal(A,Xb)) = Xb ).

% the_sym_eq_trivial
tff(fact_4313_The__split__eq,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_mb(A,fun(B,fun(A,fun(B,$o))),Xb),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y) ).

% The_split_eq
tff(fact_4314_the1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y4: A] :
              ( aa(A,$o,P,Y4)
             => ( Y4 = X4 ) ) )
     => ( aa(A,$o,P,A2)
       => ( the(A,P) = A2 ) ) ) ).

% the1_equality
tff(fact_4315_the1I2,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y4: A] :
              ( aa(A,$o,P,Y4)
             => ( Y4 = X4 ) ) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,the(A,P)) ) ) ).

% the1I2
tff(fact_4316_If__def,axiom,
    ! [A: $tType,P: $o,Xb: A,Y: A] :
      $ite((P),Xb,Y) = the(A,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_mc($o,fun(A,fun(A,fun(A,$o))),(P)),Xb),Y)) ).

% If_def
tff(fact_4317_theI2,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A2 ) )
       => ( ! [X2: A] :
              ( aa(A,$o,P,X2)
             => aa(A,$o,Q,X2) )
         => aa(A,$o,Q,the(A,P)) ) ) ) ).

% theI2
tff(fact_4318_theI_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y4: A] :
              ( aa(A,$o,P,Y4)
             => ( Y4 = X4 ) ) )
     => aa(A,$o,P,the(A,P)) ) ).

% theI'
tff(fact_4319_theI,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A2 ) )
       => aa(A,$o,P,the(A,P)) ) ) ).

% theI
tff(fact_4320_floor__real__def,axiom,
    ! [Xb: real] : archim6421214686448440834_floor(real,Xb) = the(int,aTP_Lamp_md(real,fun(int,$o),Xb)) ).

% floor_real_def
tff(fact_4321_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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),aa(num,num,bit0,one2))),aa(nat,nat,suc,Nb))),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_4322_modulo__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),Mb)),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Mb,Nb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Mb,Nb))))) ) ).

% modulo_int_unfold
tff(fact_4323_powr__int,axiom,
    ! [Xb: real,I2: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,ring_1_of_int(real,I2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,I2)),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),Xb),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I2))))) ) ) ).

% powr_int
tff(fact_4324_divide__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),Mb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),Nb))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_4325_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).

% bit_0_eq
tff(fact_4326_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_4327_sgn__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_one
tff(fact_4328_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,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),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_divide
tff(fact_4329_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sgn_sgn(A),A2)),Nb) ) ).

% power_sgn
tff(fact_4330_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,sgn_sgn(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_4331_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_greater
tff(fact_4332_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,sgn_sgn(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,sgn_sgn(A),B2)) ) ).

% divide_sgn
tff(fact_4333_nat__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_4334_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,A,sgn_sgn(A),A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_4335_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Mb))),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),Mb)),Nb) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_4336_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_4337_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Mb))),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),Mb)),Nb) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_4338_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_4339_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_4340_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% zless_nat_conj
tff(fact_4341_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K))) = ring_1_of_int(A,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_4342_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( R = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_4343_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),K))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( R = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_4344_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R))),K)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_4345_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),L)),K)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_4346_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_4347_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),zero_zero(int))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_4348_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_4349_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% zero_less_nat_eq
tff(fact_4350_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)) ) ).

% sgn_of_nat
tff(fact_4351_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: num] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,W))),aa(num,nat,numeral_numeral(nat),Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(Nb)) ) ) ).

% bit_numeral_simps(2)
tff(fact_4352_diff__nat__numeral,axiom,
    ! [V: num,V4: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V4)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V4))) ).

% diff_nat_numeral
tff(fact_4353_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W)))),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_4354_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: num] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),aa(num,nat,numeral_numeral(nat),Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(Nb)) ) ) ).

% bit_numeral_simps(3)
tff(fact_4355_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(nat,nat,suc,Nb))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_4356_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb) = aa(int,nat,nat2,Y) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_4357_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa(int,nat,nat2,Y) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb) )
    <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_4358_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,Xb))),A2)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),A2)) ) ).

% nat_ceiling_le_eq
tff(fact_4359_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),zero_zero(nat))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ).

% bit_0
tff(fact_4360_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ).

% one_less_nat_eq
tff(fact_4361_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W)))),aa(num,nat,numeral_numeral(nat),Nb))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),pred_numeral(Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_4362_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(num,nat,numeral_numeral(nat),Nb))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),pred_numeral(Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_4363_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Nb)
        <=> ( ( Nb = zero_zero(nat) )
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_4364_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_4365_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_4366_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_4367_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_4368_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_4369_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,Y: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),Xb)),aa(A,A,sgn_sgn(A),Y)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_4370_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),Mb)),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(num,nat,numeral_numeral(nat),Mb)),Nb) ) ) ).

% bit_numeral_iff
tff(fact_4371_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,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),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_mult
tff(fact_4372_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ).

% bit_and_iff
tff(fact_4373_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) )
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
              | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_4374_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A2) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,sgn_sgn(A),A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_4375_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Mb),A2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
            & ( Mb != Nb ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_4376_bit__and__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),Nb)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_4377_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),Mb)),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Mb),Nb) ) ) ).

% bit_of_nat_iff_bit
tff(fact_4378_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,Nb)) ) ).

% not_bit_1_Suc
tff(fact_4379_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),Nb)
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_4380_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% bit_numeral_simps(1)
tff(fact_4381_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),A2)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_take_bit_iff
tff(fact_4382_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: $o,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa($o,A,zero_neq_one_of_bool(A),(B2))),Nb)
        <=> ( (B2)
            & ( Nb = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_4383_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,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_4384_nat__numeral__as__int,axiom,
    ! [X4: num] : aa(num,nat,numeral_numeral(nat),X4) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X4)) ).

% nat_numeral_as_int
tff(fact_4385_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),aa(A,A,sgn_sgn(A),K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_4386_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)),aa(A,A,sgn_sgn(A),A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_4387_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,abs_abs(A),A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_4388_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),Xb)),aa(A,A,abs_abs(A),Xb)) = Xb ) ).

% mult_sgn_abs
tff(fact_4389_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ).

% nat_mono
tff(fact_4390_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N: nat,L3: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L3)),aa(nat,int,semiring_1_of_nat(int),N)) ).

% int_sgnE
tff(fact_4391_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,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_4392_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_4393_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_4394_unset__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),Mb),Nb) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Mb),aa(nat,int,semiring_1_of_nat(int),Nb))) ).

% unset_bit_nat_def
tff(fact_4395_nat__mask__eq,axiom,
    ! [Nb: nat] : aa(int,nat,nat2,bit_se2239418461657761734s_mask(int,Nb)) = bit_se2239418461657761734s_mask(nat,Nb) ).

% nat_mask_eq
tff(fact_4396_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_1_pos
tff(fact_4397_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_4398_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% nat_mono_iff
tff(fact_4399_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R)))) ) ).

% of_nat_ceiling
tff(fact_4400_zless__nat__eq__int__zless,axiom,
    ! [Mb: nat,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Mb)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_4401_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xb)),Nb)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% nat_le_iff
tff(fact_4402_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int))),Nb)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% bit_not_int_iff'
tff(fact_4403_nat__int__add,axiom,
    ! [A2: nat,B2: nat] : aa(int,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_4404_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
       => ( aa(int,int,sgn_sgn(int),modulo_modulo(int,K,L)) = aa(int,int,sgn_sgn(int),L) ) ) ) ).

% sgn_mod
tff(fact_4405_int__minus,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) = aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(nat,int,semiring_1_of_nat(int),Mb)))) ).

% int_minus
tff(fact_4406_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa(int,nat,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),aa(int,nat,nat2,aa(int,int,abs_abs(int),W))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Z))) ).

% nat_abs_mult_distrib
tff(fact_4407_real__nat__ceiling__ge,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),aa(int,nat,nat2,archimedean_ceiling(real,Xb)))) ).

% real_nat_ceiling_ge
tff(fact_4408_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          bit_se8732182000553998342ip_bit(A,Nb,A2) = aa(A,A,
            aa(nat,fun(A,A),
              $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),
              Nb),
            A2) ) ).

% flip_bit_eq_if
tff(fact_4409_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X4),Xa) = aa(int,nat,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_4410_and__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Mb),Nb) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(nat,int,semiring_1_of_nat(int),Nb))) ).

% and_nat_def
tff(fact_4411_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X4),Xa) = aa(int,nat,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_4412_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X4),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(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_4413_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X4),Xa) = aa(int,nat,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_4414_nat__mod__as__int,axiom,
    ! [X4: nat,Xa: nat] : modulo_modulo(nat,X4,Xa) = aa(int,nat,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_4415_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_4416_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          aa(A,A,sgn_sgn(A),Xb) = $ite(
            Xb = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_4417_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R)))),R) ) ) ).

% of_nat_floor
tff(fact_4418_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_4419_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),W)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_4420_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),Xb)) = $ite(Xb = zero_zero(A),zero_zero(real),one_one(real)) ) ).

% norm_sgn
tff(fact_4421_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,archim6421214686448440834_floor(A,A2))),aa(int,nat,nat2,archim6421214686448440834_floor(A,B2)))),aa(int,nat,nat2,archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ).

% le_mult_nat_floor
tff(fact_4422_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z5)
       => ( aa(int,nat,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),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_4423_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K) ) ) ).

% le_nat_iff
tff(fact_4424_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),aa(int,int,sgn_sgn(int),V)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,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_4425_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_4426_Suc__as__int,axiom,
    ! [X4: nat] : aa(nat,nat,suc,X4) = aa(int,nat,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_4427_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,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),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_4428_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z5),Z)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_4429_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_4430_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,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),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_4431_nat__div__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_4432_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,Xb)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_4433_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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),aa(int,int,sgn_sgn(int),K)),aa(int,int,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_4434_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_concat_bit(Mb,K),L)),Nb)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_4435_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(int,nat,nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_4436_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xb)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_4437_nat__mod__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,modulo_modulo(int,Xb,Y)) = modulo_modulo(nat,aa(int,nat,nat2,Xb),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_4438_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),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_4439_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_4440_le__nat__floor,axiom,
    ! [Xb: nat,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Xb)),A2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(int,nat,nat2,archim6421214686448440834_floor(real,A2))) ) ).

% le_nat_floor
tff(fact_4441_nat__take__bit__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_4442_take__bit__nat__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ) ).

% take_bit_nat_eq
tff(fact_4443_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)))))) ) ).

% divide_int_def
tff(fact_4444_modulo__int__def,axiom,
    ! [K: int,L: int] :
      modulo_modulo(int,K,L) = $ite(
        L = zero_zero(int),
        K,
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))))) ) ).

% modulo_int_def
tff(fact_4445_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = aa(int,int,bit_concat_bit(Nb,K),aa(int,int,uminus_uminus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_4446_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) = zero_zero(A) )
         => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_4447_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Nb) ) ) ).

% bit_Suc
tff(fact_4448_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
            <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_4449_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_4450_nat__2,axiom,
    aa(int,nat,nat2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_4451_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),M2)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N) ) ) ) ).

% int_bit_bound
tff(fact_4452_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_4453_nat__less__iff,axiom,
    ! [W: int,Mb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W)),Mb)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),aa(nat,int,semiring_1_of_nat(int),Mb)) ) ) ).

% nat_less_iff
tff(fact_4454_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,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),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_4455_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) ).

% nat_abs_int_diff
tff(fact_4456_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_4457_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z5)) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z5),zero_zero(int)),
        aa(int,nat,nat2,Z),
        $let(
          d: int,
          d:= aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z5),
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),d),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_4458_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),Nb))) ) ) ).

% bit_iff_odd
tff(fact_4459_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = zero_zero(A) )
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_4460_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
    <=> ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),Nb))) ) ).

% bit_int_def
tff(fact_4461_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R)),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)),R) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_4462_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_4463_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_4464_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) )
               => ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),A22) )
                 => ( aa(int,$o,aa(int,fun(int,$o),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_4465_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( 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)),K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( 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) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L4) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4))
            & ( K3 = 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_4466_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,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),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($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_4467_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,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),aa(num,num,bit0,one2))),B2))),Nb)
          <=> $ite(Nb = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)),Nb)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_4468_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> $ite(Nb = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_4469_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K) ) ) ).

% even_nat_iff
tff(fact_4470_set__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),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($o,int,zero_neq_one_of_bool(int),~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))) ).

% set_bit_eq
tff(fact_4471_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))) ).

% unset_bit_eq
tff(fact_4472_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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),aa(num,num,bit0,one2))),Nb)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_4473_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,ring_1_of_int(real,Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,Nb)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Nb))))) ) ) ).

% powr_real_of_int
tff(fact_4474_floor__rat__def,axiom,
    ! [Xb: rat] : archim6421214686448440834_floor(rat,Xb) = the(int,aTP_Lamp_me(rat,fun(int,$o),Xb)) ).

% floor_rat_def
tff(fact_4475_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X5)
     => ( finite_finite2(A,X5)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,count_list(A,Xs)),X5) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_4476_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_4477_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = $ite(
        ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
        | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) ),
        aa(int,int,uminus_uminus(int),one_one(int)),
        $ite(
          K = zero_zero(int),
          L,
          $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),ord_max(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% or_int_unfold
tff(fact_4478_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_4479_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_4480_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_4481_sgn__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sgn_sgn(real),Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_4482_zero__le__sgn__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sgn_sgn(real),Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% zero_le_sgn_iff
tff(fact_4483_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_4484_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_4485_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),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,Nb),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2)) ) ).

% take_bit_or
tff(fact_4486_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_4487_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_4488_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_4489_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_4490_count__notin,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,nat,count_list(A,Xs),Xb) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_4491_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_4492_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb)) ) ).

% or_numerals(8)
tff(fact_4493_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_4494_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),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(1)
tff(fact_4495_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb)) ) ).

% or_numerals(5)
tff(fact_4496_or__minus__numerals_I2_J,axiom,
    ! [Nb: 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,Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_4497_or__minus__numerals_I6_J,axiom,
    ! [Nb: 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,Nb)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_4498_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_4499_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_4500_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_4501_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),Nb)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
            | aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ).

% bit_or_iff
tff(fact_4502_bit__or__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),Nb)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
        | aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_4503_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_or_eq
tff(fact_4504_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R)
     => ~ ! [S2: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),S2)
           => ! [T4: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T4)
               => ( R != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S2),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_4505_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : 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),ring_1_of_int(A,K)),ring_1_of_int(A,L)) ) ).

% of_int_or_eq
tff(fact_4506_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_4507_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_4508_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_4509_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_4510_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),zero_zero(A)) = Xb ) ).

% bit.disj_zero_right
tff(fact_4511_bit_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Z)) ) ).

% bit.conj_disj_distrib
tff(fact_4512_bit_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Z)) ) ).

% bit.disj_conj_distrib
tff(fact_4513_bit_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xb: 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)),Xb) = 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),Xb)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),Xb)) ) ).

% bit.conj_disj_distrib2
tff(fact_4514_bit_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xb: 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)),Xb) = 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),Xb)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Z),Xb)) ) ).

% bit.disj_conj_distrib2
tff(fact_4515_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_4516_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_4517_OR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xb),Y)) ) ) ).

% OR_lower
tff(fact_4518_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_4519_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N: nat] :
              ( ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N)
              | ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N) )
         => ( 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_4520_plus__and__or,axiom,
    ! [Xb: 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),Xb),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xb),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Y) ).

% plus_and_or
tff(fact_4521_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : pow(Xb,one2) = Xb ).

% pow.simps(1)
tff(fact_4522_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_4523_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(Nb),Xb)) = aa(real,real,sgn_sgn(real),Xb) ) ) ).

% sgn_root
tff(fact_4524_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
            & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2) ) ) ) ).

% even_or_iff
tff(fact_4525_sgn__real__def,axiom,
    ! [A2: real] :
      aa(real,real,sgn_sgn(real),A2) = $ite(
        A2 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_4526_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Xb) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Xb) = 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)) )
               => ( Xb = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_4527_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(int,nat,nat2,K)),Nb)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ) ).

% bit_nat_iff
tff(fact_4528_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),Xb)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% count_le_length
tff(fact_4529_sgn__power__injE,axiom,
    ! [A2: real,Nb: nat,Xb: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A2)),Nb)) = Xb )
     => ( ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),Nb)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_4530_bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Mb),Nb)
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))) ) ).

% bit_nat_def
tff(fact_4531_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,Nb)) = 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),aa(num,num,bit0,one2))),Nb)),bit_se2239418461657761734s_mask(A,Nb)) ) ).

% mask_Suc_exp
tff(fact_4532_root__sgn__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_4533_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),aa(real,real,root(Nb),Xb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),aa(real,real,root(Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_4534_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(Xb) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => ( arg(Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_4535_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))) ) ).

% or_one_eq
tff(fact_4536_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))) ) ).

% one_or_eq
tff(fact_4537_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,Nb)) = 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),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,Nb))) ) ).

% mask_Suc_double
tff(fact_4538_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xb),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ) ) ).

% OR_upper
tff(fact_4539_split__root,axiom,
    ! [P: fun(real,$o),Nb: nat,Xb: real] :
      ( aa(real,$o,P,aa(real,real,root(Nb),Xb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ! [Y3: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y3)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y3)),Nb)) = Xb )
             => aa(real,$o,P,Y3) ) ) ) ) ).

% split_root
tff(fact_4540_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(arg(Z)) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_4541_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($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K)
            | ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% or_int_rec
tff(fact_4542_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero(real) )
     => ( aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xb)) = aa(real,real,aa(real,fun(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),aa(real,real,sgn_sgn(real),Xb)),pi)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(real,real,arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_4543_or__minus__numerals_I5_J,axiom,
    ! [Nb: 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,bit0,Nb)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Nb)))) ).

% or_minus_numerals(5)
tff(fact_4544_or__minus__numerals_I1_J,axiom,
    ! [Nb: 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,bit0,Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Nb)))) ).

% or_minus_numerals(1)
tff(fact_4545_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_mf(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_4546_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( aa(set(real),$o,member(real,Nb),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),aa(num,num,bit0,one2))),pi)),Nb)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_4547_some__sym__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : fChoice(A,fequal(A,Xb)) = Xb ).

% some_sym_eq_trivial
tff(fact_4548_some__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : fChoice(A,aTP_Lamp_ek(A,fun(A,$o),Xb)) = Xb ).

% some_eq_trivial
tff(fact_4549_some__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A2 ) )
       => ( fChoice(A,P) = A2 ) ) ) ).

% some_equality
tff(fact_4550_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(set(A),$o,member(A,Xb),ring_1_Ints(A))
            | aa(set(A),$o,member(A,Y),ring_1_Ints(A)) )
         => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_4551_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),archimedean_frac(A,Xb))
        <=> ~ aa(set(A),$o,member(A,Xb),ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_4552_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb)) ).

% or_nat_numerals(4)
tff(fact_4553_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_4554_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb)) ).

% or_nat_numerals(3)
tff(fact_4555_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),aa(num,num,bit0,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_4556_or__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Mb,aa(num,num,bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_4557_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: 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,Nb)))),aa(num,int,numeral_numeral(int),Mb)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Mb,aa(num,num,bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_4558_or__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Mb,bitM(Nb)))) ).

% or_minus_numerals(3)
tff(fact_4559_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: 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,bit0,Nb)))),aa(num,int,numeral_numeral(int),Mb)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Mb,bitM(Nb)))) ).

% or_minus_numerals(7)
tff(fact_4560_diff__rat__def,axiom,
    ! [Q2: rat,R: rat] : aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q2),R) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q2),aa(rat,rat,uminus_uminus(rat),R)) ).

% diff_rat_def
tff(fact_4561_some__in__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3))),A3)
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_4562_some1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y4: A] :
              ( aa(A,$o,P,Y4)
             => ( Y4 = X4 ) ) )
     => ( aa(A,$o,P,A2)
       => ( fChoice(A,P) = A2 ) ) ) ).

% some1_equality
tff(fact_4563_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
    <=> ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% some_eq_ex
tff(fact_4564_someI2__bex,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),A3)
          & aa(A,$o,P,X4) )
     => ( ! [X2: A] :
            ( ( aa(set(A),$o,member(A,X2),A3)
              & aa(A,$o,P,X2) )
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_co(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) ) ) ).

% someI2_bex
tff(fact_4565_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X_12: A] : aa(A,$o,P,X_12)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2_ex
tff(fact_4566_someI__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X_12: A] : aa(A,$o,P,X_12)
     => aa(A,$o,P,fChoice(A,P)) ) ).

% someI_ex
tff(fact_4567_someI2,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2
tff(fact_4568_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_4569_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,B2),ring_1_Ints(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_4570_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: num] : aa(set(A),$o,member(A,aa(num,A,numeral_numeral(A),Nb)),ring_1_Ints(A)) ) ).

% Ints_numeral
tff(fact_4571_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_fy(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
          <=> aa(A,$o,P3,X2) )
       => ( ! [X_1: A] : aa(A,$o,P3,X_1)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_4572_verit__sko__forall__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_fy(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_1: A] : aa(A,$o,P,X_1)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_forall_indirect
tff(fact_4573_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( Xb = fChoice(A,P) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
          <=> aa(A,$o,P3,X2) )
       => ( ? [X_1: A] : aa(A,$o,P3,X_1)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_4574_verit__sko__ex__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,P) )
     => ( ? [X_1: A] : aa(A,$o,P,X_1)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_ex_indirect
tff(fact_4575_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B4: A,A3: A,P: fun(A,$o)] :
      ( ( B4 = A3 )
     => ( ( fChoice(A,P) = A3 )
      <=> ( fChoice(A,P) = B4 ) ) ) ).

% verit_sko_forall''
tff(fact_4576_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_fy(fun(A,$o),fun(A,$o),P)))
      <=> (A3) )
     => ( ! [X_1: A] : aa(A,$o,P,X_1)
      <=> (A3) ) ) ).

% verit_sko_forall'
tff(fact_4577_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_1: A] : aa(A,$o,P,X_1)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_fy(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_4578_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A3) )
     => ( ? [X_1: A] : aa(A,$o,P,X_1)
      <=> (A3) ) ) ).

% verit_sko_ex'
tff(fact_4579_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => aa(set(A),$o,member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_4580_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,B2),ring_1_Ints(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),ring_1_Ints(A)) ) ) ) ).

% Ints_add
tff(fact_4581_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(set(A),$o,member(A,one_one(A)),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_4582_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,B2),ring_1_Ints(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),ring_1_Ints(A)) ) ) ) ).

% Ints_diff
tff(fact_4583_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( aa(set(A),$o,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_4584_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : bit_or_not_num_neg(aa(num,num,bit0,Nb),one2) = aa(num,num,bit0,one2) ).

% or_not_num_neg.simps(4)
tff(fact_4585_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,Mb: num] : bit_or_not_num_neg(aa(num,num,bit0,Nb),aa(num,num,bit1,Mb)) = aa(num,num,bit0,bit_or_not_num_neg(Nb,Mb)) ).

% or_not_num_neg.simps(6)
tff(fact_4586_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : bit_or_not_num_neg(aa(num,num,bit1,Nb),one2) = one2 ).

% or_not_num_neg.simps(7)
tff(fact_4587_or__not__num__neg_Osimps_I3_J,axiom,
    ! [Mb: num] : bit_or_not_num_neg(one2,aa(num,num,bit1,Mb)) = aa(num,num,bit1,Mb) ).

% or_not_num_neg.simps(3)
tff(fact_4588_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_mg(A,fun(A,fun(A,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_4589_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,Mb: num] : bit_or_not_num_neg(aa(num,num,bit0,Nb),aa(num,num,bit0,Mb)) = bitM(bit_or_not_num_neg(Nb,Mb)) ).

% or_not_num_neg.simps(5)
tff(fact_4590_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,Mb: num] : bit_or_not_num_neg(aa(num,num,bit1,Nb),aa(num,num,bit1,Mb)) = bitM(bit_or_not_num_neg(Nb,Mb)) ).

% or_not_num_neg.simps(9)
tff(fact_4591_or__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Mb),Nb) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(nat,int,semiring_1_of_nat(int),Nb))) ).

% or_nat_def
tff(fact_4592_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( aa(set(A),$o,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_4593_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),B2),A2)
         => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),ring_1_of_int(A,A2)),ring_1_of_int(A,B2))),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_4594_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_mh(A,fun(A,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_4595_or__not__num__neg_Osimps_I2_J,axiom,
    ! [Mb: num] : bit_or_not_num_neg(one2,aa(num,num,bit0,Mb)) = aa(num,num,bit1,Mb) ).

% or_not_num_neg.simps(2)
tff(fact_4596_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,Mb: num] : bit_or_not_num_neg(aa(num,num,bit1,Nb),aa(num,num,bit0,Mb)) = bitM(bit_or_not_num_neg(Nb,Mb)) ).

% or_not_num_neg.simps(8)
tff(fact_4597_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),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))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_4598_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(set(A),$o,member(A,Xb),ring_1_Ints(A))
         => ( ( Xb != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),Xb)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_4599_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(set(A),$o,member(A,Xb),ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xb)),one_one(A))
           => ( Xb = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_4600_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,member(A,Xb),ring_1_Ints(A))
         => ( aa(set(A),$o,member(A,Y),ring_1_Ints(A))
           => ( ( Xb = Y )
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_4601_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),pi)) = zero_zero(real) )
    <=> aa(set(real),$o,member(real,Xb),ring_1_Ints(real)) ) ).

% sin_times_pi_eq_0
tff(fact_4602_or__not__num__neg_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( bit_or_not_num_neg(Xb,Xaa) = Y )
     => ( ( ( Xb = one2 )
         => ( ( Xaa = one2 )
           => ( Y != one2 ) ) )
       => ( ( ( Xb = one2 )
           => ! [M: num] :
                ( ( Xaa = aa(num,num,bit0,M) )
               => ( Y != aa(num,num,bit1,M) ) ) )
         => ( ( ( Xb = one2 )
             => ! [M: num] :
                  ( ( Xaa = aa(num,num,bit1,M) )
                 => ( Y != aa(num,num,bit1,M) ) ) )
           => ( ( ? [N: num] : Xb = aa(num,num,bit0,N)
               => ( ( Xaa = one2 )
                 => ( Y != aa(num,num,bit0,one2) ) ) )
             => ( ! [N: num] :
                    ( ( Xb = aa(num,num,bit0,N) )
                   => ! [M: num] :
                        ( ( Xaa = aa(num,num,bit0,M) )
                       => ( Y != bitM(bit_or_not_num_neg(N,M)) ) ) )
               => ( ! [N: num] :
                      ( ( Xb = aa(num,num,bit0,N) )
                     => ! [M: num] :
                          ( ( Xaa = aa(num,num,bit1,M) )
                         => ( Y != aa(num,num,bit0,bit_or_not_num_neg(N,M)) ) ) )
                 => ( ( ? [N: num] : Xb = aa(num,num,bit1,N)
                     => ( ( Xaa = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N: num] :
                          ( ( Xb = aa(num,num,bit1,N) )
                         => ! [M: num] :
                              ( ( Xaa = aa(num,num,bit0,M) )
                             => ( Y != bitM(bit_or_not_num_neg(N,M)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xb = aa(num,num,bit1,N) )
                           => ! [M: num] :
                                ( ( Xaa = aa(num,num,bit1,M) )
                               => ( Y != bitM(bit_or_not_num_neg(N,M)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_4603_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),Xb)) = $ite(aa(set(A),$o,member(A,Xb),ring_1_Ints(A)),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,Xb))) ) ).

% frac_neg
tff(fact_4604_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: A] :
          ( ( archimedean_frac(A,Xb) = A2 )
        <=> ( aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2)),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_4605_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),ring_1_of_int(B,aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2)))),ring_1_of_int(B,archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_4606_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(set(A),$o,member(A,A2),ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),ring_1_of_int(B,archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))),ring_1_of_int(B,aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_4607_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa(set(real),$o,member(real,Nb),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),aa(num,num,bit0,one2))),pi)),Nb)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_4608_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa(set(real),$o,member(real,Nb),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),aa(num,num,bit0,one2))),pi)),Nb)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_4609_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))) ).

% Suc_0_or_eq
tff(fact_4610_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))) ).

% or_Suc_0_eq
tff(fact_4611_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Mb),Nb) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb)
            | ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% or_nat_rec
tff(fact_4612_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Mb),Nb) = $ite(
        Mb = zero_zero(nat),
        Nb,
        $ite(Nb = zero_zero(nat),Mb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% or_nat_unfold
tff(fact_4613_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_mi(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_4614_setceilmax,axiom,
    ! [S3: vEBT_VEBT,Mb: nat,Listy: list(vEBT_VEBT),Nb: nat] :
      ( vEBT_invar_vebt(S3,Mb)
     => ( ! [X2: vEBT_VEBT] :
            ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
           => vEBT_invar_vebt(X2,Nb) )
       => ( ( Mb = aa(nat,nat,suc,Nb) )
         => ( ! [X2: vEBT_VEBT] :
                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X2),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,X2)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Nb))) ) )
           => ( ( 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),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Mb))) )
             => ( aa(nat,int,semiring_1_of_nat(int),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),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),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),Mb))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_4615_case__prod__app,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F2: fun(B,fun(C,fun(D,A))),Xb: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F2),Xb),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_mj(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F2),Y)),Xb) ).

% case_prod_app
tff(fact_4616_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_mk(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_4617_image__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ml(A,A)),Y6) = Y6 ).

% image_ident
tff(fact_4618_height__compose__list,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,Ta),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summarya),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))))) ) ).

% height_compose_list
tff(fact_4619_max__ins__scaled,axiom,
    ! [Nb: nat,X14: vEBT_VEBT,Mb: nat,X13: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(nat),set(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_4620_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_mb(A,fun(B,fun(A,fun(B,$o))),Xb),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y) ).

% Eps_case_prod_eq
tff(fact_4621_height__i__max,axiom,
    ! [I2: nat,X13: list(vEBT_VEBT),Foo: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Foo),aa(set(nat),nat,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_4622_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_4623_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or1337092689740270186AtMost(A,I2,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost
tff(fact_4624_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D2),A2)) ) ).

% image_diff_atLeastAtMost
tff(fact_4625_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)),aa(A,set(A),set_ord_atMost(A),A2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)) ) ).

% image_add_atMost
tff(fact_4626_max__idx__list,axiom,
    ! [I2: nat,X13: list(vEBT_VEBT),Nb: nat,X14: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I2)))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(nat),nat,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_4627_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_mm(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I2,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_4628_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_mn(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_4629_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bh(nat,fun(nat,$o),Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_4630_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xb)
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xb) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_4631_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xb)
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xb) ) ) ) ) ) ).

% Max_less_iff
tff(fact_4632_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A3: set(A),C2: B] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,aTP_Lamp_mo(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).

% Max_const
tff(fact_4633_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4634_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_mp(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4635_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S: set(A),F2: fun(A,B),K: B] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_mq(fun(A,B),fun(B,fun(A,B)),F2),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,F2),S))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_4636_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,F2),A3))
     => ~ ! [X2: B] :
            ( ( B2 = aa(B,A,F2,X2) )
           => ~ aa(set(B),$o,member(B,X2),A3) ) ) ).

% imageE
tff(fact_4637_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_mr(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A3) ).

% image_image
tff(fact_4638_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: set(B),P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_ms(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F2),A3),P)) = aa(set(B),set(A),image(B,A,F2),aa(fun(B,$o),set(B),collect(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_mt(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F2),A3),P))) ).

% Compr_image_eq
tff(fact_4639_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( ~ finite_finite2(A,A3)
     => ( finite_finite2(B,aa(set(A),set(B),image(A,B,F2),A3))
       => ? [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
            & ~ finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_mu(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A3),F2),X2))) ) ) ) ).

% pigeonhole_infinite
tff(fact_4640_divide__rat__def,axiom,
    ! [Q2: rat,R: rat] : aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),Q2),R) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Q2),aa(rat,rat,inverse_inverse(rat),R)) ).

% divide_rat_def
tff(fact_4641_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B),B4: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(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),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A3),B4))) ).

% image_diff_subset
tff(fact_4642_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,Xb),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ).

% Max_ge
tff(fact_4643_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ! [Y4: A] :
                ( aa(set(A),$o,member(A,Y4),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
           => ( aa(set(A),$o,member(A,Xb),A3)
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = Xb ) ) ) ) ) ).

% Max_eqI
tff(fact_4644_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),A3)
                 => ? [Xa: A] :
                      ( aa(set(A),$o,member(A,Xa),B4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa) ) )
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),B4)
                   => ? [Xa: A] :
                        ( aa(set(A),$o,member(A,Xa),A3)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,lattic643756798349783984er_Max(A),B4) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_4645_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ).

% Max.coboundedI
tff(fact_4646_image__constant,axiom,
    ! [A: $tType,B: $tType,Xb: A,A3: set(A),C2: B] :
      ( aa(set(A),$o,member(A,Xb),A3)
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_mv(B,fun(A,B),C2)),A3) = aa(set(B),set(B),insert(B,C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_4647_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A3: set(B)] :
      aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_mw(A,fun(B,A)),C2)),A3) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),insert(A,C2),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_4648_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S) = aa(set(C),B,groups7311177749621191930dd_sum(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_my(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H),G)),aa(set(A),set(C),image(A,C,G),S)) ) ) ) ).

% sum.image_gen
tff(fact_4649_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_mz(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H),G)),aa(set(A),set(C),image(A,C,G),S)) ) ) ) ).

% prod.image_gen
tff(fact_4650_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_na(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).

% split_paired_Eps
tff(fact_4651_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Mb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = Mb )
            <=> ( aa(set(A),$o,member(A,Mb),A3)
                & ! [X: A] :
                    ( aa(set(A),$o,member(A,X),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Mb) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_4652_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A3))
            <=> ? [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_4653_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Mb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( Mb = aa(set(A),A,lattic643756798349783984er_Max(A),A3) )
            <=> ( aa(set(A),$o,member(A,Mb),A3)
                & ! [X: A] :
                    ( aa(set(A),$o,member(A,X),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Mb) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_4654_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xb)
             => ! [A8: A] :
                  ( aa(set(A),$o,member(A,A8),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A8),Xb) ) ) ) ) ) ).

% Max.boundedE
tff(fact_4655_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xb) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),Xb) ) ) ) ) ).

% Max.boundedI
tff(fact_4656_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A3))
            <=> ? [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_4657_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( ! [B3: A] :
                ( aa(set(A),$o,member(A,B3),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A2) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,A2),A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_4658_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Uu,Dega,TreeLista,Summarya)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summarya),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))))) ).

% VEBT_internal.height.simps(2)
tff(fact_4659_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(A),T2: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( finite_finite2(B,T2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S)),T2)
             => ( aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_nc(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H)),T2) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,H),S) ) ) ) ) ) ).

% sum.group
tff(fact_4660_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(A),T2: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( finite_finite2(B,T2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S)),T2)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_nd(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H)),T2) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),H),S) ) ) ) ) ) ).

% prod.group
tff(fact_4661_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ finite_finite2(A,A3)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_4662_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),Xb),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_4663_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B4)) ) ) ) ) ).

% Max.subset_imp
tff(fact_4664_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M6),N2)
         => ( ( M6 != bot_bot(set(A)) )
           => ( finite_finite2(A,N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M6)),aa(set(A),A,lattic643756798349783984er_Max(A),N2)) ) ) ) ) ).

% Max_mono
tff(fact_4665_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xb) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Uu2,Deg,TreeList,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_4666_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) = $ite(Nb = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ne(nat,fun(nat,fun(nat,$o)),Mb),Nb)))) ).

% divide_nat_def
tff(fact_4667_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_nf(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_4668_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ng(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_abs_code
tff(fact_4669_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F2: fun(fun(A,B),C),G: C] :
          ( ! [X2: fun(A,B)] : aa(fun(A,B),C,F2,X2) = G
         => ( aa(fun(A,B),C,F2,aTP_Lamp_nh(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_4670_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,Xb),A3)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_4671_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,Xb),A3)
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_4672_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),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),Xb)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4673_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P2),Q2)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_nj(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_4674_rat__floor__code,axiom,
    ! [P2: rat] : archim6421214686448440834_floor(rat,P2) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),divide_divide(int)),quotient_of(P2)) ).

% rat_floor_code
tff(fact_4675_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          aa(set(A),set(A),image(A,A,aTP_Lamp_nk(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)),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),Xb),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4676_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P2),Q2)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_nm(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_4677_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_nn(A,fun(A,fun(A,A)),Mb),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),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),Mb),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C2)),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),Mb),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4678_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_no(A,fun(A,fun(A,A)),Mb),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4679_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_np(A,fun(A,fun(A,A)),Mb),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),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),Mb)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Mb)),C2)),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),Mb)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Mb)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4680_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_nq(A,fun(A,fun(A,A)),Mb),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Mb),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Mb)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Mb)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Mb)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Mb)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4681_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(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_4682_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_4683_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),A2)),aa(set(A),set(A),uminus_uminus(set(A)),Ta)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),A2)),Ta)) ) ).

% translation_subtract_Compl
tff(fact_4684_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S3: set(A),Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),A2)),S3)),aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),A2)),Ta)) ) ).

% translation_subtract_diff
tff(fact_4685_bij__betw__Suc,axiom,
    ! [M6: set(nat),N2: set(nat)] :
      ( bij_betw(nat,nat,suc,M6,N2)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M6) = N2 ) ) ).

% bij_betw_Suc
tff(fact_4686_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,I2,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I2),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_4687_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set(product_prod(A,B)),F2: fun(A,fun(B,C))] :
      ( aa(set(product_prod(A,B)),$o,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)
     => aa(set(C),$o,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,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2)),A3)) ) ).

% pair_imageI
tff(fact_4688_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,$o),set(int),collect(int),aTP_Lamp_bg(int,fun(int,$o),Nb))) = aa(int,int,abs_abs(int),Nb) ) ) ).

% Max_divisors_self_int
tff(fact_4689_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ aa(set(nat),$o,member(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),A3)) ).

% zero_notin_Suc_image
tff(fact_4690_None__notin__image__Some,axiom,
    ! [A: $tType,A3: set(A)] : ~ aa(set(option(A)),$o,member(option(A),none(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),A3)) ).

% None_notin_image_Some
tff(fact_4691_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
    <=> ? [N4: nat,F5: fun(nat,A)] : A3 = aa(set(nat),set(A),image(nat,A,F5),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),N4))) ) ).

% finite_conv_nat_seg_image
tff(fact_4692_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F2: fun(nat,A),Nb: nat] :
      ( ( A3 = aa(set(nat),set(A),image(nat,A,F2),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),Nb))) )
     => finite_finite2(A,A3) ) ).

% nat_seg_image_imp_finite
tff(fact_4693_in__image__insert__iff,axiom,
    ! [A: $tType,B4: set(set(A)),Xb: A,A3: set(A)] :
      ( ! [C7: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),C7),B4)
         => ~ aa(set(A),$o,member(A,Xb),C7) )
     => ( aa(set(set(A)),$o,member(set(A),A3),aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,Xb)),B4))
      <=> ( aa(set(A),$o,member(A,Xb),A3)
          & aa(set(set(A)),$o,member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B4) ) ) ) ).

% in_image_insert_iff
tff(fact_4694_image__Suc__lessThan,axiom,
    ! [Nb: nat] : aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),Nb) ).

% image_Suc_lessThan
tff(fact_4695_image__Suc__atMost,axiom,
    ! [Nb: nat] : aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,Nb)) ).

% image_Suc_atMost
tff(fact_4696_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_4697_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_4698_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_4699_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S3: set(A),Ta: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),Ta)) = aa(set(A),set(A),aa(set(A),fun(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)),Ta)) ) ).

% translation_diff
tff(fact_4700_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: 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)),Ta)) = 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)),Ta)) ) ).

% translation_Compl
tff(fact_4701_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_nt(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_4702_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),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_nv(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_4703_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),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_nx(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_4704_normalize__crossproduct,axiom,
    ! [Q2: int,S3: int,P2: int,R: 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),R),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),R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_4705_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),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_nz(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_4706_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))) ).

% xor_Suc_0_eq
tff(fact_4707_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))) ).

% Suc_0_xor_eq
tff(fact_4708_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(list($o),nat,size_size(list($o)),Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_4709_bit_Oxor__left__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Y)) = Y ) ).

% bit.xor_left_self
tff(fact_4710_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Xb) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_4711_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_4712_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_4713_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_4714_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),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,Nb),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2)) ) ).

% take_bit_xor
tff(fact_4715_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_4716_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb)) ) ).

% xor_numerals(8)
tff(fact_4717_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb)) ) ).

% xor_numerals(5)
tff(fact_4718_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),aa(num,num,bit0,Y)) ) ).

% xor_numerals(2)
tff(fact_4719_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),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_4720_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_4721_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Xb)) ).

% xor_nat_numerals(4)
tff(fact_4722_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb)) ).

% xor_nat_numerals(3)
tff(fact_4723_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),aa(num,num,bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_4724_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),aa(num,num,bit0,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_4725_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: 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,Xb))),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_4726_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Xb))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_4727_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),Nb)
        <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
            <=> aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),Nb) ) ) ) ).

% bit_xor_iff
tff(fact_4728_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : 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),ring_1_of_int(A,K)),ring_1_of_int(A,L)) ) ).

% of_int_xor_eq
tff(fact_4729_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_4730_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_4731_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_4732_bit_Oconj__xor__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xb: 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)),Xb) = 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),Xb)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),Xb)) ) ).

% bit.conj_xor_distrib2
tff(fact_4733_bit_Oconj__xor__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Z)) ) ).

% bit.conj_xor_distrib
tff(fact_4734_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),Mb)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_xor_eq
tff(fact_4735_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2) ) ) ) ).

% even_xor_iff
tff(fact_4736_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Mb),Nb) = $ite(
        Mb = zero_zero(nat),
        Nb,
        $ite(Nb = zero_zero(nat),Mb,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,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% xor_nat_unfold
tff(fact_4737_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Mb),Nb) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb) != ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),Mb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% xor_nat_rec
tff(fact_4738_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))) ) ).

% one_xor_eq
tff(fact_4739_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2))) ) ).

% xor_one_eq
tff(fact_4740_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),Bs)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),Nb) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_4741_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),Nb),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),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),Nb))) ) ).

% push_bit_numeral_minus_1
tff(fact_4742_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F2: fun(A,B),I2: A] :
          ( finite_finite2(A,F4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_oa(set(A),fun(fun(A,B),fun(A,$o)),I5),F2))),F4)
           => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,I2),I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I2)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_4743_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_height_rel,Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = 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)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,Deg,TreeList,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Node(Uu2,Deg,TreeList,Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_4744_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se4730199178511100633sh_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_4745_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se4730199178511100633sh_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_4746_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( bit_se4730199178511100633sh_bit(A,Nb,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_4747_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se4730199178511100633sh_bit(A,Nb,zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_4748_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Mb,bit_se4730199178511100633sh_bit(A,Nb,A2)) = bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb),A2) ) ).

% push_bit_push_bit
tff(fact_4749_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Nb,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,Nb,A2)),bit_se4730199178511100633sh_bit(A,Nb,B2)) ) ).

% push_bit_and
tff(fact_4750_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Nb,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,Nb,A2)),bit_se4730199178511100633sh_bit(A,Nb,B2)) ) ).

% push_bit_or
tff(fact_4751_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Nb,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,Nb,A2)),bit_se4730199178511100633sh_bit(A,Nb,B2)) ) ).

% push_bit_xor
tff(fact_4752_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa(int,int,bit_concat_bit(Nb,zero_zero(int)),L) = bit_se4730199178511100633sh_bit(int,Nb,L) ).

% concat_bit_of_zero_1
tff(fact_4753_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_4754_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_4755_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),K)) = bit_se4730199178511100633sh_bit(A,Nb,aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) ) ).

% push_bit_Suc_numeral
tff(fact_4756_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: num] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_4757_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),aa(num,num,bit0,K))) ) ).

% push_bit_numeral
tff(fact_4758_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P2: fun(A,B),I2: A] :
          ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),P2)))
         => ( groups1027152243600224163dd_sum(A,B,P2,aa(set(A),set(A),insert(A,I2),I5)) = $ite(aa(set(A),$o,member(A,I2),I5),groups1027152243600224163dd_sum(A,B,P2,I5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P2,I2)),groups1027152243600224163dd_sum(A,B,P2,I5))) ) ) ) ).

% sum.insert'
tff(fact_4759_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : bit_se4730199178511100633sh_bit(nat,Nb,aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb) ).

% push_bit_of_Suc_0
tff(fact_4760_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb),A2) = bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% push_bit_Suc
tff(fact_4761_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se4730199178511100633sh_bit(A,Nb,one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) ) ).

% push_bit_of_1
tff(fact_4762_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se4730199178511100633sh_bit(A,Nb,A2))
        <=> ( ( Nb != zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_4763_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),aa(num,num,bit0,K)))) ) ).

% push_bit_minus_numeral
tff(fact_4764_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),Nb)
    <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_4765_flip__bit__int__def,axiom,
    ! [Nb: nat,K: int] : bit_se8732182000553998342ip_bit(int,Nb,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),bit_se4730199178511100633sh_bit(int,Nb,one_one(int))) ).

% flip_bit_int_def
tff(fact_4766_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4730199178511100633sh_bit(A,Nb,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,Nb,A2)),bit_se4730199178511100633sh_bit(A,Nb,B2)) ) ).

% push_bit_add
tff(fact_4767_push__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : bit_se4730199178511100633sh_bit(nat,Nb,aa(int,nat,nat2,K)) = aa(int,nat,nat2,bit_se4730199178511100633sh_bit(int,Nb,K)) ).

% push_bit_nat_eq
tff(fact_4768_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se4730199178511100633sh_bit(nat,Mb,Nb)) = bit_se4730199178511100633sh_bit(A,Mb,aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_push_bit
tff(fact_4769_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat] : bit_se4730199178511100633sh_bit(A,Nb,aa(nat,A,semiring_1_of_nat(A),Mb)) = aa(nat,A,semiring_1_of_nat(A),bit_se4730199178511100633sh_bit(nat,Nb,Mb)) ) ).

% push_bit_of_nat
tff(fact_4770_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: int] : bit_se4730199178511100633sh_bit(A,Nb,ring_1_of_int(A,K)) = ring_1_of_int(A,bit_se4730199178511100633sh_bit(int,Nb,K)) ) ).

% push_bit_of_int
tff(fact_4771_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),bit_se4730199178511100633sh_bit(A,Nb,A2)) ) ).

% push_bit_minus
tff(fact_4772_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ob(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_4773_XOR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xb),Y)) ) ) ).

% XOR_lower
tff(fact_4774_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Mb,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),bit_se4730199178511100633sh_bit(A,Mb,A2)) ) ).

% push_bit_take_bit
tff(fact_4775_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),bit_se4730199178511100633sh_bit(A,Nb,A2)) = bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),A2)) ) ).

% take_bit_push_bit
tff(fact_4776_set__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),Mb),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Nb),bit_se4730199178511100633sh_bit(nat,Mb,one_one(nat))) ).

% set_bit_nat_def
tff(fact_4777_flip__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : bit_se8732182000553998342ip_bit(nat,Mb,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Nb),bit_se4730199178511100633sh_bit(nat,Mb,one_one(nat))) ).

% flip_bit_nat_def
tff(fact_4778_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oc(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_4779_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,bit_se4730199178511100633sh_bit(int,Mb,K)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_4780_xor__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Mb),Nb) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(nat,int,semiring_1_of_nat(int),Nb))) ).

% xor_nat_def
tff(fact_4781_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,bit_se4730199178511100633sh_bit(nat,Mb,Q2)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_4782_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_concat_bit(Nb,K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),bit_se4730199178511100633sh_bit(int,Nb,L)) ).

% concat_bit_eq
tff(fact_4783_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Nb),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),bit_se4730199178511100633sh_bit(A,Nb,one_one(A))) ) ).

% set_bit_eq_or
tff(fact_4784_concat__bit__def,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_concat_bit(Nb,K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),bit_se4730199178511100633sh_bit(int,Nb,L)) ).

% concat_bit_def
tff(fact_4785_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se8732182000553998342ip_bit(A,Nb,A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),bit_se4730199178511100633sh_bit(A,Nb,one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_4786_set__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),bit_se4730199178511100633sh_bit(int,Nb,one_one(int))) ).

% set_bit_int_def
tff(fact_4787_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,G,X2) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S) = groups1027152243600224163dd_sum(A,B,G,T2) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_4788_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,G,X2) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T2) = groups1027152243600224163dd_sum(A,B,G,S) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_4789_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,H,I3) = zero_zero(B) ) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S)
                 => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,S) = groups1027152243600224163dd_sum(A,B,H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_4790_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,G,X2) = zero_zero(B) ) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S)
                 => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T2) = groups1027152243600224163dd_sum(A,B,H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_4791_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oc(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_4792_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A),I5: set(B)] :
          groups1027152243600224163dd_sum(B,A,P2,I5) = $ite(finite_finite2(B,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ob(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,P2),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ob(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_4793_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),bit_se4730199178511100633sh_bit(A,Nb,A2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% push_bit_double
tff(fact_4794_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se4730199178511100633sh_bit(A,Nb,one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_4795_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : bit_se4730199178511100633sh_bit(int,Nb,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),aa(num,num,bit0,one2))),Nb)) ).

% push_bit_int_def
tff(fact_4796_push__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : bit_se4730199178511100633sh_bit(nat,Nb,Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% push_bit_nat_def
tff(fact_4797_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Nb,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),aa(num,num,bit0,one2))),Nb)) ) ).

% push_bit_eq_mult
tff(fact_4798_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)),A2)
         => ~ ! [B3: A] : A2 != bit_se4730199178511100633sh_bit(A,Nb,B3) ) ) ).

% exp_dvdE
tff(fact_4799_push__bit__minus__one,axiom,
    ! [Nb: nat] : bit_se4730199178511100633sh_bit(int,Nb,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),aa(num,num,bit0,one2))),Nb)) ).

% push_bit_minus_one
tff(fact_4800_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xb),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_4801_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),I2: A] :
          ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_oa(set(A),fun(fun(A,B),fun(A,$o)),I5),F2)))
         => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert(A,I2),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,I2),I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I2)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ).

% sum_diff1'
tff(fact_4802_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = $let(
            l: A,
            l:= aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A2),
            $ite(aa(nat,$o,bit_se5641148757651400278ts_bit(A,l),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_4803_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($o,int,zero_neq_one_of_bool(int),
            ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K) != ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% xor_int_rec
tff(fact_4804_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_maxt_rel,Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      aa(nat,option(nat),some(nat),one_one(nat)),
                      $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = 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)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = 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)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma) )
                   => ~ 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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_4805_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_mint_rel,Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = $ite(
                      (A4),
                      aa(nat,option(nat),some(nat),zero_zero(nat)),
                      $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = 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)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = 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)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi) )
                   => ~ 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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_4806_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((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] :
                ( ( Xb = 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)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = 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)),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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_4807_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((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] :
                ( ( Xb = 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)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = 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)),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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_4808_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,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one(nat) )
             => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = 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),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) )
                     => ( ( 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),Va2,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_4809_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
        K = aa(int,int,uminus_uminus(int),one_one(int)),
        aa(int,int,bit_ri4277139882892585799ns_not(int),L),
        $ite(
          L = aa(int,int,uminus_uminus(int),one_one(int)),
          aa(int,int,bit_ri4277139882892585799ns_not(int),K),
          $ite(
            K = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_4810_bit_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) = aa(A,A,bit_ri4277139882892585799ns_not(A),Y) )
        <=> ( Xb = Y ) ) ) ).

% bit.compl_eq_compl_iff
tff(fact_4811_bit_Odouble__compl,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = Xb ) ).

% bit.double_compl
tff(fact_4812_bit_Oxor__compl__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),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),Xb),Y)) ) ).

% bit.xor_compl_right
tff(fact_4813_bit_Oxor__compl__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Y) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Y)) ) ).

% bit.xor_compl_left
tff(fact_4814_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_4815_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_4816_bit_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_disj
tff(fact_4817_bit_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_conj
tff(fact_4818_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_4819_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_4820_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_4821_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_4822_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_4823_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_4824_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) ) ).

% bit.xor_one_right
tff(fact_4825_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) ) ).

% bit.xor_one_left
tff(fact_4826_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_4827_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_4828_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),inc(Nb)) ) ).

% minus_not_numeral_eq
tff(fact_4829_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4277139882892585799ns_not(A),A2))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A2) ) ) ).

% even_not_iff
tff(fact_4830_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_4831_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),aa(num,num,bit0,one2))) ) ) ).

% not_one_eq
tff(fact_4832_or__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: 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),Mb))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Mb)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Nb)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_4833_and__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: 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),Mb))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Mb)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Nb)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_4834_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2) ) ) ) ).

% take_bit_not_iff
tff(fact_4835_take__bit__not__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) ) ).

% take_bit_not_take_bit
tff(fact_4836_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : 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_4837_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] : ring_1_of_int(A,aa(int,int,bit_ri4277139882892585799ns_not(int),K)) = aa(A,A,bit_ri4277139882892585799ns_not(A),ring_1_of_int(A,K)) ) ).

% of_int_not_eq
tff(fact_4838_bit__not__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),Nb)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Nb) ) ).

% bit_not_int_iff
tff(fact_4839_not__diff__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),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_4840_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,aa(A,fun(A,A),minus_minus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_add_distrib
tff(fact_4841_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_4842_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_4843_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_4844_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_4845_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,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_4846_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),one_one(A)) ) ).

% not_eq_complement
tff(fact_4847_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_4848_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_4849_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_4850_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A2: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,B2),N)
             => aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),N) )
         => ( aa(A,A,aa(A,fun(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_4851_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),bit_se2239418461657761734s_mask(A,Nb)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_4852_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% minus_numeral_inc_eq
tff(fact_4853_bit_Oxor__def2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),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),Xb),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))) ) ).

% bit.xor_def2
tff(fact_4854_bit_Oxor__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),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),Xb),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),Xb)),Y)) ) ).

% bit.xor_def
tff(fact_4855_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_4856_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),aa(num,num,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),aa(num,num,bit0,one2)))) ).

% not_int_div_2
tff(fact_4857_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K) ) ).

% even_not_iff_int
tff(fact_4858_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb))) ) ).

% not_numeral_Bit0_eq
tff(fact_4859_and__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,Mb)) ).

% and_not_numerals(4)
tff(fact_4860_and__not__numerals_I2_J,axiom,
    ! [Nb: 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,bit0,Nb)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_4861_or__not__numerals_I2_J,axiom,
    ! [Nb: 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,bit0,Nb)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Nb))) ).

% or_not_numerals(2)
tff(fact_4862_or__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Mb))),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_4863_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K)),Nb)
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))),Nb) ) ).

% bit_minus_int_iff
tff(fact_4864_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb))) ) ).

% not_numeral_BitM_eq
tff(fact_4865_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_4866_int__numeral__or__not__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Mb,Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_4867_int__numeral__not__or__num__neg,axiom,
    ! [Mb: num,Nb: 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),Mb))),aa(num,int,numeral_numeral(int),Nb)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Nb,Mb))) ).

% int_numeral_not_or_num_neg
tff(fact_4868_numeral__or__not__num__eq,axiom,
    ! [Mb: num,Nb: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Mb,Nb)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb)))) ).

% numeral_or_not_num_eq
tff(fact_4869_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat] : bit_se4730199178511100633sh_bit(A,Mb,bit_se2239418461657761734s_mask(A,Nb)) = 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),Nb),Mb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Mb))) ) ).

% push_bit_mask_eq
tff(fact_4870_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Nb),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,Nb,one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_4871_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),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,Nb,one_one(int)))) ).

% unset_bit_int_def
tff(fact_4872_and__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Nb)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb)))) ).

% and_not_numerals(5)
tff(fact_4873_and__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,Mb)) ).

% and_not_numerals(7)
tff(fact_4874_or__not__numerals_I3_J,axiom,
    ! [Nb: 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,Nb)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Nb))) ).

% or_not_numerals(3)
tff(fact_4875_and__not__numerals_I3_J,axiom,
    ! [Nb: 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,Nb)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_4876_or__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Mb))),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_4877_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_4878_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_4879_and__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb)))) ).

% and_not_numerals(6)
tff(fact_4880_and__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb)))) ).

% and_not_numerals(9)
tff(fact_4881_or__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb)))) ).

% or_not_numerals(6)
tff(fact_4882_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),Nb)
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) != zero_zero(A) )
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ) ).

% bit_not_iff_eq
tff(fact_4883_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)) ) ).

% minus_exp_eq_not_mask
tff(fact_4884_or__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Nb)))) = 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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))))) ).

% or_not_numerals(5)
tff(fact_4885_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)))) ) ).

% signed_take_bit_def
tff(fact_4886_and__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Nb)))) = 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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))))) ).

% and_not_numerals(8)
tff(fact_4887_or__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb)))) = 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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))))) ).

% or_not_numerals(9)
tff(fact_4888_or__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Mb))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Nb)))) = 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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))))) ).

% or_not_numerals(8)
tff(fact_4889_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($o,int,zero_neq_one_of_bool(int),aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2)))))) ).

% not_int_rec
tff(fact_4890_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) )
                     => ( ~ (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),Va2,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_4891_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_4892_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = 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_4893_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ! [Uv2: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv2)) )
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2) )
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va2,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_4894_Cauchy__iff2,axiom,
    ! [X5: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X5)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M5)
         => ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,X5,M5)),aa(nat,real,X5,N4)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_4895_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_df(nat,nat)),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Ico_nat
tff(fact_4896_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% dual_order.refl
tff(fact_4897_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Xb) ) ).

% order_refl
tff(fact_4898_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I2),set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_4899_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_4900_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,J: A,Mb: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I2,J)),set_or7035219750837199246ssThan(A,Mb,Nb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),I2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),I2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Nb) ) ) ) ) ).

% ivl_subset
tff(fact_4901_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4902_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_4903_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_4904_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or7035219750837199246ssThan(A,I2,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan
tff(fact_4905_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,Nb: A,Mb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),Nb)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I2,Mb)),set_or7035219750837199246ssThan(A,I2,Nb)) = set_or7035219750837199246ssThan(A,Nb,Mb) ) ) ) ).

% ivl_diff
tff(fact_4906_lessThan__minus__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Nb: A,Mb: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(A,set(A),set_ord_lessThan(A),Nb)),aa(A,set(A),set_ord_lessThan(A),Mb)) = set_or7035219750837199246ssThan(A,Mb,Nb) ) ).

% lessThan_minus_lessThan
tff(fact_4907_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,I2,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I2),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_4908_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_mm(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I2,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_4909_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Mb)) = aa(set(nat),set(nat),insert(nat,Mb),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_4910_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.op_ivl_Suc
tff(fact_4911_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.op_ivl_Suc
tff(fact_4912_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4913_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4914_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),D2)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
            <=> ( ( A2 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4915_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4916_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_4917_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Nb)
          & aa(nat,$o,P,M5) )
    <=> ? [X: nat] :
          ( aa(set(nat),$o,member(nat,X),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less_eq
tff(fact_4918_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Nb)
         => aa(nat,$o,P,M5) )
    <=> ! [X: nat] :
          ( aa(set(nat),$o,member(nat,X),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less_eq
tff(fact_4919_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_4920_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cv(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_4921_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cw(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_4922_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_4923_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hy(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_4924_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),D2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_4925_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),D2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_4926_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P2)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Nb,P2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_4927_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,P2: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P2)
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,Mb,P2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,Mb,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,Nb,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_4928_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,P2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_4929_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_4930_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_4931_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4932_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ) ).

% order_class.order_eq_iff
tff(fact_4933_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_4934_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_4935_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
           => ( Xb = Y ) ) ) ) ).

% order_antisym
tff(fact_4936_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% order.trans
tff(fact_4937_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) ) ) ) ).

% order_trans
tff(fact_4938_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
         => ( ! [A4: A,B3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B3),A4)
               => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
           => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ).

% linorder_wlog
tff(fact_4939_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.eq_iff
tff(fact_4940_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_4941_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% dual_order.trans
tff(fact_4942_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_4943_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funD
tff(fact_4944_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funE
tff(fact_4945_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G) ) ) ).

% le_funI
tff(fact_4946_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
        <=> ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ) ).

% le_fun_def
tff(fact_4947_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_4948_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X2: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X2)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_subst1
tff(fact_4949_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X2: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_subst2
tff(fact_4950_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% order_eq_refl
tff(fact_4951_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_linear
tff(fact_4952_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X2: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X2)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_4953_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X2: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_4954_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_le_cases
tff(fact_4955_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_4956_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xb: A] :
        ? [Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xb) ) ).

% lt_ex
tff(fact_4957_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xb: A] :
        ? [X_13: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X_13) ) ).

% gt_ex
tff(fact_4958_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [Z4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z4)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),Y) ) ) ) ).

% dense
tff(fact_4959_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( Xb != Y ) ) ) ).

% less_imp_neq
tff(fact_4960_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order.asym
tff(fact_4961_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_4962_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_4963_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [X2: A] :
              ( ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X2)
                 => aa(A,$o,P,Y5) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A2) ) ) ).

% less_induct
tff(fact_4964_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv3
tff(fact_4965_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ( Xb != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_cases
tff(fact_4966_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% dual_order.asym
tff(fact_4967_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% dual_order.irrefl
tff(fact_4968_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P,X_1)
        <=> ? [N4: A] :
              ( aa(A,$o,P,N4)
              & ! [M5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M5),N4)
                 => ~ aa(A,$o,P,M5) ) ) ) ) ).

% exists_least_iff
tff(fact_4969_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P,A4),A4)
           => ( ! [A4: A,B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B3),A4)
                 => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_4970_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans
tff(fact_4971_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
            | ( Xb = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4972_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans
tff(fact_4973_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_4974_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4975_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_neqE
tff(fact_4976_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% order_less_asym
tff(fact_4977_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ) ).

% linorder_neq_iff
tff(fact_4978_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order_less_asym'
tff(fact_4979_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_less_trans
tff(fact_4980_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F2,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X2: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X2)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_4981_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( aa(A,B,F2,B2) = C2 )
           => ( ! [X2: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_4982_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Xb) ) ).

% order_less_irrefl
tff(fact_4983_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X2: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X2)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_4984_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X2: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_4985_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% order_less_not_sym
tff(fact_4986_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_4987_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
          | ( Xb = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% linorder_less_linear
tff(fact_4988_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( Xb != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_4989_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( Y != Xb ) ) ) ).

% order_less_imp_not_eq2
tff(fact_4990_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% order_less_imp_not_less
tff(fact_4991_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4992_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4993_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_4994_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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),Nb))),aa(nat,A,G,Nb)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_4995_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_4996_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),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_4997_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4998_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4999_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5000_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_5001_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Nb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% sum.last_plus
tff(fact_5002_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% prod.last_plus
tff(fact_5003_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_db(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,Nb)),aa(nat,A,F2,Mb)) ) ) ) ).

% sum_Suc_diff'
tff(fact_5004_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      set_or7035219750837199246ssThan(nat,Mb,aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb),aa(set(nat),set(nat),insert(nat,Nb),set_or7035219750837199246ssThan(nat,Mb,Nb)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_5005_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_od(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% sum.atLeastLessThan_rev
tff(fact_5006_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_oe(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iv(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% sum.nested_swap
tff(fact_5007_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_5008_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_of(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% prod.atLeastLessThan_rev
tff(fact_5009_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_og(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_iy(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% prod.nested_swap
tff(fact_5010_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_oh(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% sum.nat_group
tff(fact_5011_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_oi(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% prod.nat_group
tff(fact_5012_prod__Suc__fact,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) = semiring_char_0_fact(nat,Nb) ).

% prod_Suc_fact
tff(fact_5013_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = semiring_char_0_fact(nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_5014_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.head_if
tff(fact_5015_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.head_if
tff(fact_5016_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% fact_prod_Suc
tff(fact_5017_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_cy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_5018_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Nb,Mb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ic(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_5019_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,Nb) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_if(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_prod
tff(fact_5020_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K: num] :
      set_or7035219750837199246ssThan(nat,Mb,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),pred_numeral(K)),aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_or7035219750837199246ssThan(nat,Mb,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_5021_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aa(nat,fun(nat,nat),minus_minus(nat),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% fact_prod_rev
tff(fact_5022_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [N5: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),M5)
                 => ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F2),set_or7035219750837199246ssThan(nat,M5,N4)))),E4) ) ) ) ) ).

% summable_Cauchy
tff(fact_5023_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M8: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M5)
                 => ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N4)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,M5)),aa(nat,A,X5,N4)))),E4) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_5024_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M9: nat] :
                ! [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,M)),aa(nat,A,X5,N)))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI
tff(fact_5025_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => ? [M7: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M2)
               => ! [N8: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N8)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,M2)),aa(nat,A,X5,N8)))),E) ) ) ) ) ) ).

% CauchyD
tff(fact_5026_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A,K: nat] :
          ( aa(A,$o,sums(A,F2),S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_oj(fun(nat,A),fun(nat,fun(nat,A)),F2),K)),S3) ) ) ) ).

% sums_group
tff(fact_5027_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ok(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% take_bit_sum
tff(fact_5028_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% leD
tff(fact_5029_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% leI
tff(fact_5030_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_5031_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv1
tff(fact_5032_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv2
tff(fact_5033_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_ge
tff(fact_5034_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_le
tff(fact_5035_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ) ).

% less_le_not_le
tff(fact_5036_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ).

% not_le_imp_less
tff(fact_5037_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_5038_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_5039_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans1
tff(fact_5040_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% order.strict_trans2
tff(fact_5041_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% order.strict_iff_not
tff(fact_5042_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
         => ( ! [W2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),W2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Xb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),W2) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_5043_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( ! [W2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),W2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_5044_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_5045_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_5046_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_5047_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_5048_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_5049_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% order.strict_implies_order
tff(fact_5050_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% dual_order.strict_implies_order
tff(fact_5051_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( Xb = Y ) ) ) ) ).

% order_le_less
tff(fact_5052_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & ( Xb != Y ) ) ) ) ).

% order_less_le
tff(fact_5053_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% linorder_not_le
tff(fact_5054_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_not_less
tff(fact_5055_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% order_less_imp_le
tff(fact_5056_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( A2 != B2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_5057_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_5058_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_le_less_trans
tff(fact_5059_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z) ) ) ) ).

% order_less_le_trans
tff(fact_5060_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C2)
           => ( ! [X2: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X2)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_5061_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),C2)
           => ( ! [X2: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_5062_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F2: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X2: B,Y4: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X2)),aa(B,A,F2,Y4)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_5063_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F2: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B2)),C2)
           => ( ! [X2: A,Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_5064_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb) ) ) ).

% linorder_le_less_linear
tff(fact_5065_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
            | ( Xb = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_5066_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A2) ) ).

% bot.extremum
tff(fact_5067_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_5068_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_5069_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_5070_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( ( A2 != bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A2) ) ) ).

% bot.not_eq_extremum
tff(fact_5071_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xb: nat,Y: nat] :
      aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_ol(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,Xb,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_5072_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_lessThan(nat),Nb)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_5073_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_5074_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y) = Xb ) ) ) ).

% max_absorb1
tff(fact_5075_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),B2,A2) ) ).

% max_def
tff(fact_5076_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( semiring_char_0_fact(A,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K),Nb)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K))) ) ) ) ).

% fact_split
tff(fact_5077_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_om(nat,fun(nat,fun(nat,A)),K),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_5078_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_on(A,fun(nat,fun(nat,A)),A2),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_5079_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)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_oo(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_5080_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)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_oo(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_5081_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),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_lg(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_5082_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),aa(num,num,bit0,one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_5083_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_op(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_5084_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I3: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,I3)),aa(nat,A,A2,J2)) ) )
         => ( ! [I3: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I3)) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_oq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))),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),Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_5085_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I3: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,A2,I3)),aa(nat,nat,A2,J2)) ) )
     => ( ! [I3: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,B2,J2)),aa(nat,nat,B2,I3)) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_or(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))),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),Nb))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_5086_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_5087_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_os(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_5088_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_5089_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(A,B),B4: set(B)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
         => aa(set(B),$o,member(B,aa(A,B,F2,X2)),B4) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),aa(fun(A,$o),set(A),collect(A),P))),B4) ) ).

% image_Collect_subsetI
tff(fact_5090_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
     => vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq2
tff(fact_5091_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
    <=> vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq
tff(fact_5092_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(Ta,D2)
     => vEBT_VEBT_valid(Ta,D2) ) ).

% valid_eq1
tff(fact_5093_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D2: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv)),D2)
    <=> ( D2 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_5094_size__list__estimation,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(A,nat,F2,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),size_list(A,F2,Xs)) ) ) ).

% size_list_estimation
tff(fact_5095_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X2)),aa(A,nat,G,X2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),size_list(A,F2,Xs)),size_list(A,G,Xs)) ) ).

% size_list_pointwise
tff(fact_5096_size__list__estimation_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F2: fun(A,nat)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F2,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),size_list(A,F2,Xs)) ) ) ).

% size_list_estimation'
tff(fact_5097_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R: A,S3: B,R2: set(product_prod(A,B)),S6: B] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S3)),R2)
     => ( ( S6 = S3 )
       => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S6)),R2) ) ) ).

% ssubst_Pair_rhs
tff(fact_5098_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_5099_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),aa(num,num,bit0,one2))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_subseqs
tff(fact_5100_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : re(csqrt(Z)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% csqrt.simps(1)
tff(fact_5101_complex__Re__numeral,axiom,
    ! [V: num] : re(aa(num,complex,numeral_numeral(complex),V)) = aa(num,real,numeral_numeral(real),V) ).

% complex_Re_numeral
tff(fact_5102_Re__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),S3: set(A)] : re(aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,F2),S3)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_ot(fun(A,complex),fun(A,real),F2)),S3) ).

% Re_sum
tff(fact_5103_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(num,complex,numeral_numeral(complex),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Z)),aa(num,real,numeral_numeral(real),W)) ).

% Re_divide_numeral
tff(fact_5104_sums__Re,axiom,
    ! [X5: fun(nat,complex),A2: complex] :
      ( aa(complex,$o,sums(complex,X5),A2)
     => aa(real,$o,sums(real,aTP_Lamp_ou(fun(nat,complex),fun(nat,real),X5)),re(A2)) ) ).

% sums_Re
tff(fact_5105_Cauchy__Re,axiom,
    ! [X5: fun(nat,complex)] :
      ( topolo3814608138187158403Cauchy(complex,X5)
     => topolo3814608138187158403Cauchy(real,aTP_Lamp_ou(fun(nat,complex),fun(nat,real),X5)) ) ).

% Cauchy_Re
tff(fact_5106_complex__Re__le__cmod,axiom,
    ! [Xb: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(Xb)),real_V7770717601297561774m_norm(complex,Xb)) ).

% complex_Re_le_cmod
tff(fact_5107_one__complex_Osimps_I1_J,axiom,
    re(one_one(complex)) = one_one(real) ).

% one_complex.simps(1)
tff(fact_5108_plus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),re(Xb)),re(Y)) ).

% plus_complex.simps(1)
tff(fact_5109_scaleR__complex_Osimps_I1_J,axiom,
    ! [R: real,Xb: complex] : re(aa(complex,complex,real_V8093663219630862766scaleR(complex,R),Xb)) = aa(real,real,aa(real,fun(real,real),times_times(real),R),re(Xb)) ).

% scaleR_complex.simps(1)
tff(fact_5110_minus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),Xb),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),re(Xb)),re(Y)) ).

% minus_complex.simps(1)
tff(fact_5111_summable__Re,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
     => summable(real,aTP_Lamp_ou(fun(nat,complex),fun(nat,real),F2)) ) ).

% summable_Re
tff(fact_5112_abs__Re__le__cmod,axiom,
    ! [Xb: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),re(Xb))),real_V7770717601297561774m_norm(complex,Xb)) ).

% abs_Re_le_cmod
tff(fact_5113_Re__csqrt,axiom,
    ! [Z: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(csqrt(Z))) ).

% Re_csqrt
tff(fact_5114_subset__subseqs,axiom,
    ! [A: $tType,X5: set(A),Xs: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(list(A),set(A),set2(A),Xs))
     => aa(set(set(A)),$o,member(set(A),X5),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_5115_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),zero_zero(real))
    <=> ( re(Z) = aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_5116_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A2: real] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A2)) = re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_5117_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_5118_csqrt_Ocode,axiom,
    ! [Z: complex] :
      csqrt(Z) = complex2(aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),
        aa(real,real,
          aa(real,fun(real,real),times_times(real),
            $ite(im(Z) = zero_zero(real),one_one(real),aa(real,real,sgn_sgn(real),im(Z)))),
          aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% csqrt.code
tff(fact_5119_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      im(csqrt(Z)) = aa(real,real,
        aa(real,fun(real,real),times_times(real),
          $ite(im(Z) = zero_zero(real),one_one(real),aa(real,real,sgn_sgn(real),im(Z)))),
        aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% csqrt.simps(2)
tff(fact_5120_csqrt__of__real__nonpos,axiom,
    ! [Xb: complex] :
      ( ( im(Xb) = zero_zero(real) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(Xb)),zero_zero(real))
       => ( csqrt(Xb) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,aa(real,real,abs_abs(real),re(Xb))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_5121_Im__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),S3: set(A)] : im(aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,F2),S3)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_ov(fun(A,complex),fun(A,real),F2)),S3) ).

% Im_sum
tff(fact_5122_Im__i__times,axiom,
    ! [Z: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z)) = re(Z) ).

% Im_i_times
tff(fact_5123_Re__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( im(Xb) = zero_zero(real) )
     => ( re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xb),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),Nb) ) ) ).

% Re_power_real
tff(fact_5124_Re__i__times,axiom,
    ! [Z: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z)) = aa(real,real,uminus_uminus(real),im(Z)) ).

% Re_i_times
tff(fact_5125_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(num,complex,numeral_numeral(complex),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),aa(num,real,numeral_numeral(real),W)) ).

% Im_divide_numeral
tff(fact_5126_csqrt__of__real__nonneg,axiom,
    ! [Xb: complex] :
      ( ( im(Xb) = zero_zero(real) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(Xb))
       => ( csqrt(Xb) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,re(Xb))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_5127_csqrt__minus,axiom,
    ! [Xb: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(Xb)),zero_zero(real))
        | ( ( im(Xb) = zero_zero(real) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(Xb)) ) )
     => ( csqrt(aa(complex,complex,uminus_uminus(complex),Xb)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),csqrt(Xb)) ) ) ).

% csqrt_minus
tff(fact_5128_sums__Im,axiom,
    ! [X5: fun(nat,complex),A2: complex] :
      ( aa(complex,$o,sums(complex,X5),A2)
     => aa(real,$o,sums(real,aTP_Lamp_ow(fun(nat,complex),fun(nat,real),X5)),im(A2)) ) ).

% sums_Im
tff(fact_5129_Cauchy__Im,axiom,
    ! [X5: fun(nat,complex)] :
      ( topolo3814608138187158403Cauchy(complex,X5)
     => topolo3814608138187158403Cauchy(real,aTP_Lamp_ow(fun(nat,complex),fun(nat,real),X5)) ) ).

% Cauchy_Im
tff(fact_5130_imaginary__unit_Osimps_I2_J,axiom,
    im(imaginary_unit) = one_one(real) ).

% imaginary_unit.simps(2)
tff(fact_5131_plus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),im(Xb)),im(Y)) ).

% plus_complex.simps(2)
tff(fact_5132_scaleR__complex_Osimps_I2_J,axiom,
    ! [R: real,Xb: complex] : im(aa(complex,complex,real_V8093663219630862766scaleR(complex,R),Xb)) = aa(real,real,aa(real,fun(real,real),times_times(real),R),im(Xb)) ).

% scaleR_complex.simps(2)
tff(fact_5133_minus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),Xb),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),im(Xb)),im(Y)) ).

% minus_complex.simps(2)
tff(fact_5134_sums__complex__iff,axiom,
    ! [F2: fun(nat,complex),Xb: complex] :
      ( aa(complex,$o,sums(complex,F2),Xb)
    <=> ( aa(real,$o,sums(real,aTP_Lamp_ou(fun(nat,complex),fun(nat,real),F2)),re(Xb))
        & aa(real,$o,sums(real,aTP_Lamp_ow(fun(nat,complex),fun(nat,real),F2)),im(Xb)) ) ) ).

% sums_complex_iff
tff(fact_5135_summable__Im,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
     => summable(real,aTP_Lamp_ow(fun(nat,complex),fun(nat,real),F2)) ) ).

% summable_Im
tff(fact_5136_abs__Im__le__cmod,axiom,
    ! [Xb: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),im(Xb))),real_V7770717601297561774m_norm(complex,Xb)) ).

% abs_Im_le_cmod
tff(fact_5137_summable__complex__iff,axiom,
    ! [F2: fun(nat,complex)] :
      ( summable(complex,F2)
    <=> ( summable(real,aTP_Lamp_ou(fun(nat,complex),fun(nat,real),F2))
        & summable(real,aTP_Lamp_ow(fun(nat,complex),fun(nat,real),F2)) ) ) ).

% summable_complex_iff
tff(fact_5138_times__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),im(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),re(Y))) ).

% times_complex.simps(2)
tff(fact_5139_cmod__Re__le__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( im(Xb) = im(Y) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Xb)),real_V7770717601297561774m_norm(complex,Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),re(Xb))),aa(real,real,abs_abs(real),re(Y))) ) ) ).

% cmod_Re_le_iff
tff(fact_5140_cmod__Im__le__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( re(Xb) = re(Y) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Xb)),real_V7770717601297561774m_norm(complex,Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),im(Xb))),aa(real,real,abs_abs(real),im(Y))) ) ) ).

% cmod_Im_le_iff
tff(fact_5141_times__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xb),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),im(Y))) ).

% times_complex.simps(1)
tff(fact_5142_plus__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Xb),Y) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),re(Xb)),re(Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),im(Xb)),im(Y))) ).

% plus_complex.code
tff(fact_5143_scaleR__complex_Ocode,axiom,
    ! [R: real,Xb: complex] : aa(complex,complex,real_V8093663219630862766scaleR(complex,R),Xb) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R),re(Xb)),aa(real,real,aa(real,fun(real,real),times_times(real),R),im(Xb))) ).

% scaleR_complex.code
tff(fact_5144_minus__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),Xb),Y) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),re(Xb)),re(Y)),aa(real,real,aa(real,fun(real,real),minus_minus(real),im(Xb)),im(Y))) ).

% minus_complex.code
tff(fact_5145_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(csqrt(Z)))
      | ( ( re(csqrt(Z)) = zero_zero(real) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(csqrt(Z))) ) ) ).

% csqrt_principal
tff(fact_5146_cmod__le,axiom,
    ! [Z: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z)))) ).

% cmod_le
tff(fact_5147_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A2: real] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A2)) = im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_5148_Re__exp,axiom,
    ! [Z: complex] : re(aa(complex,complex,exp(complex),Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,exp(real),re(Z))),cos(real,im(Z))) ).

% Re_exp
tff(fact_5149_Im__exp,axiom,
    ! [Z: complex] : im(aa(complex,complex,exp(complex),Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,exp(real),re(Z))),sin(real,im(Z))) ).

% Im_exp
tff(fact_5150_fun__complex__eq,axiom,
    ! [A: $tType,F2: fun(A,complex),X4: A] : aa(A,complex,F2,X4) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),re(aa(A,complex,F2,X4)))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),im(aa(A,complex,F2,X4))))) ).

% fun_complex_eq
tff(fact_5151_complex__eq,axiom,
    ! [A2: complex] : A2 = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(real,complex,real_Vector_of_real(complex),re(A2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),im(A2)))) ).

% complex_eq
tff(fact_5152_times__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xb),Y) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),im(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),re(Y)))) ).

% times_complex.code
tff(fact_5153_exp__eq__polar,axiom,
    ! [Z: complex] : aa(complex,complex,exp(complex),Z) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),aa(real,real,exp(real),re(Z)))),cis(im(Z))) ).

% exp_eq_polar
tff(fact_5154_cmod__power2,axiom,
    ! [Z: complex] : aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% cmod_power2
tff(fact_5155_Im__power2,axiom,
    ! [Xb: complex] : im(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = 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),aa(num,num,bit0,one2))),re(Xb))),im(Xb)) ).

% Im_power2
tff(fact_5156_Re__power2,axiom,
    ! [Xb: complex] : re(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% Re_power2
tff(fact_5157_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero(complex) )
    <=> ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(real) ) ) ).

% complex_eq_0
tff(fact_5158_norm__complex__def,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,Z) = 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),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% norm_complex_def
tff(fact_5159_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : re(aa(complex,complex,inverse_inverse(complex),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% inverse_complex.simps(1)
tff(fact_5160_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% complex_neq_0
tff(fact_5161_Re__divide,axiom,
    ! [Xb: complex,Y: complex] : re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),Y)) = 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,aa(real,fun(real,real),times_times(real),re(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Re_divide
tff(fact_5162_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),W),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Z )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(W))
          | ( ( re(W) = zero_zero(real) )
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(W)) ) )
       => ( csqrt(Z) = W ) ) ) ).

% csqrt_unique
tff(fact_5163_csqrt__square,axiom,
    ! [B2: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(B2))
        | ( ( re(B2) = zero_zero(real) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(B2)) ) )
     => ( csqrt(aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = B2 ) ) ).

% csqrt_square
tff(fact_5164_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : im(aa(complex,complex,inverse_inverse(complex),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),im(Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% inverse_complex.simps(2)
tff(fact_5165_Im__divide,axiom,
    ! [Xb: complex,Y: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Im_divide
tff(fact_5166_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),real_V7770717601297561774m_norm(complex,Z))) ).

% complex_abs_le_norm
tff(fact_5167_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( 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),divide_divide(real),re(Z)),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),im(Z)),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) ) ) ).

% complex_unit_circle
tff(fact_5168_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : aa(complex,complex,inverse_inverse(complex),Xb) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),re(Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),im(Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% inverse_complex.code
tff(fact_5169_Complex__divide,axiom,
    ! [Xb: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),Y) = complex2(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,aa(real,fun(real,real),times_times(real),re(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),im(Xb)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(Xb)),im(Y)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% Complex_divide
tff(fact_5170_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ! [X2: list(A)] :
          ( aa(set(list(A)),$o,member(list(A),X2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X2) = Nb ) )
     => ( 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)),Nb) ) ) ).

% length_mul_elem
tff(fact_5171_Im__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( aa(set(complex),$o,member(complex,R),real_Vector_Reals(complex))
     => ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R),Z)) = 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,uminus_uminus(real),re(R))),im(Z))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% Im_Reals_divide
tff(fact_5172_Re__Reals__divide,axiom,
    ! [R: complex,Z: complex] :
      ( aa(set(complex),$o,member(complex,R),real_Vector_Reals(complex))
     => ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),R),Z)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(R)),re(Z))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% Re_Reals_divide
tff(fact_5173_real__eq__imaginary__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( aa(set(complex),$o,member(complex,Y),real_Vector_Reals(complex))
     => ( aa(set(complex),$o,member(complex,Xb),real_Vector_Reals(complex))
       => ( ( Xb = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y) )
        <=> ( ( Xb = zero_zero(complex) )
            & ( Y = zero_zero(complex) ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_5174_imaginary__eq__real__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( aa(set(complex),$o,member(complex,Y),real_Vector_Reals(complex))
     => ( aa(set(complex),$o,member(complex,Xb),real_Vector_Reals(complex))
       => ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Y) = Xb )
        <=> ( ( Xb = zero_zero(complex) )
            & ( Y = zero_zero(complex) ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_5175_Reals__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),real_Vector_Reals(A))
         => ( aa(set(A),$o,member(A,B2),real_Vector_Reals(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),real_Vector_Reals(A)) ) ) ) ).

% Reals_diff
tff(fact_5176_Reals__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,Nb: nat] :
          ( aa(set(A),$o,member(A,A2),real_Vector_Reals(A))
         => aa(set(A),$o,member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),real_Vector_Reals(A)) ) ) ).

% Reals_power
tff(fact_5177_Reals__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => aa(set(A),$o,member(A,one_one(A)),real_Vector_Reals(A)) ) ).

% Reals_1
tff(fact_5178_Reals__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : aa(set(A),$o,member(A,aa(num,A,numeral_numeral(A),W)),real_Vector_Reals(A)) ) ).

% Reals_numeral
tff(fact_5179_Reals__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),real_Vector_Reals(A))
         => ( aa(set(A),$o,member(A,B2),real_Vector_Reals(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),real_Vector_Reals(A)) ) ) ) ).

% Reals_add
tff(fact_5180_Reals__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),real_Vector_Reals(A))
         => ( aa(set(A),$o,member(A,B2),real_Vector_Reals(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),real_Vector_Reals(A)) ) ) ) ).

% Reals_mult
tff(fact_5181_Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),real_Vector_Reals(A))
         => ( aa(set(A),$o,member(A,B2),real_Vector_Reals(A))
           => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),real_Vector_Reals(A)) ) ) ) ).

% Reals_divide
tff(fact_5182_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,member(A,A2),real_Vector_Reals(A))
         => ( aa(set(A),$o,member(A,B2),real_Vector_Reals(A))
           => ( ( B2 != zero_zero(A) )
             => aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),real_Vector_Reals(A)) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_5183_Re__prod__Reals,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,complex)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => aa(set(complex),$o,member(complex,aa(A,complex,F2,X2)),real_Vector_Reals(complex)) )
     => ( re(aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7121269368397514597t_prod(A,complex),F2),A3)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),aTP_Lamp_ot(fun(A,complex),fun(A,real),F2)),A3) ) ) ).

% Re_prod_Reals
tff(fact_5184_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,complex),N2: nat,F2: fun(nat,A)] :
          ( summable(complex,G)
         => ( ! [N: nat] : aa(set(complex),$o,member(complex,aa(nat,complex,G,N)),real_Vector_Reals(complex))
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(nat,complex,G,N)))
             => ( ! [N: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),real_V7770717601297561774m_norm(complex,aa(nat,complex,G,N))) )
               => summable(A,F2) ) ) ) ) ) ).

% series_comparison_complex
tff(fact_5185_complex__diff__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),Z),cnj(Z)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),im(Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_5186_set__n__lists,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),$o),aTP_Lamp_ox(nat,fun(list(A),fun(list(A),$o)),Nb),Xs)) ).

% set_n_lists
tff(fact_5187_complex__mult__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% complex_mult_cnj
tff(fact_5188_complex__cnj__mult,axiom,
    ! [Xb: complex,Y: complex] : cnj(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xb),Y)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cnj(Xb)),cnj(Y)) ).

% complex_cnj_mult
tff(fact_5189_complex__cnj__add,axiom,
    ! [Xb: complex,Y: complex] : cnj(aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Xb),Y)) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),cnj(Xb)),cnj(Y)) ).

% complex_cnj_add
tff(fact_5190_complex__cnj__diff,axiom,
    ! [Xb: complex,Y: complex] : cnj(aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),Xb),Y)) = aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),cnj(Xb)),cnj(Y)) ).

% complex_cnj_diff
tff(fact_5191_cnj__sum,axiom,
    ! [A: $tType,F2: fun(A,complex),S3: set(A)] : cnj(aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,F2),S3)) = aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_oy(fun(A,complex),fun(A,complex),F2)),S3) ).

% cnj_sum
tff(fact_5192_cnj__prod,axiom,
    ! [A: $tType,F2: fun(A,complex),S3: set(A)] : cnj(aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7121269368397514597t_prod(A,complex),F2),S3)) = aa(set(A),complex,aa(fun(A,complex),fun(set(A),complex),groups7121269368397514597t_prod(A,complex),aTP_Lamp_oy(fun(A,complex),fun(A,complex),F2)),S3) ).

% cnj_prod
tff(fact_5193_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z))) = zero_zero(real) ).

% complex_In_mult_cnj_zero
tff(fact_5194_sums__cnj,axiom,
    ! [F2: fun(nat,complex),L: complex] :
      ( aa(complex,$o,sums(complex,aTP_Lamp_oz(fun(nat,complex),fun(nat,complex),F2)),cnj(L))
    <=> aa(complex,$o,sums(complex,F2),L) ) ).

% sums_cnj
tff(fact_5195_Re__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)) = zero_zero(real) )
    <=> ( re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))) = zero_zero(real) ) ) ).

% Re_complex_div_eq_0
tff(fact_5196_Im__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)) = zero_zero(real) )
    <=> ( im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))) = zero_zero(real) ) ) ).

% Im_complex_div_eq_0
tff(fact_5197_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,Z) = aa(real,real,sqrt,re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_5198_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),Nb: nat,Xs: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)))
     => ( aa(list(A),nat,size_size(list(A)),Ys) = Nb ) ) ).

% length_n_lists_elem
tff(fact_5199_Re__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Re_complex_div_lt_0
tff(fact_5200_Re__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Re_complex_div_gt_0
tff(fact_5201_Re__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Re_complex_div_ge_0
tff(fact_5202_Re__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Re_complex_div_le_0
tff(fact_5203_Im__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Im_complex_div_lt_0
tff(fact_5204_Im__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Im_complex_div_gt_0
tff(fact_5205_Im__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ).

% Im_complex_div_ge_0
tff(fact_5206_Im__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real)) ) ).

% Im_complex_div_le_0
tff(fact_5207_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% complex_mod_mult_cnj
tff(fact_5208_complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) )
      & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2)))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))) ) ) ).

% complex_div_gt_0
tff(fact_5209_length__n__lists,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,Nb,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ).

% length_n_lists
tff(fact_5210_complex__norm__square,axiom,
    ! [Z: complex] : aa(real,complex,real_Vector_of_real(complex),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) ).

% complex_norm_square
tff(fact_5211_complex__add__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Z),cnj(Z)) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),re(Z))) ).

% complex_add_cnj
tff(fact_5212_complex__div__cnj,axiom,
    ! [A2: complex,B2: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),A2),B2) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))),aa(real,complex,real_Vector_of_real(complex),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(complex,B2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% complex_div_cnj
tff(fact_5213_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cnj(Z)),W)) = aa(real,complex,real_Vector_of_real(complex),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))))) ).

% cnj_add_mult_eq_Re
tff(fact_5214_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),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(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_pa(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_5215_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_pb(set(A),fun(fun(A,B),fun(A,$o)),A3),F2)))) ) ) ) ).

% even_sum_iff
tff(fact_5216_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,A),V: num,Nb: nat] : case_nat(A,A2,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),Nb)) ).

% case_nat_add_eq_if
tff(fact_5217_card__Collect__less__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_5218_card__atMost,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_atMost(nat),U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_5219_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or7035219750837199246ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_atLeastLessThan
tff(fact_5220_card__Collect__le__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_al(nat,fun(nat,$o)),Nb))) = aa(nat,nat,suc,Nb) ).

% card_Collect_le_nat
tff(fact_5221_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,U)),L) ).

% card_atLeastAtMost
tff(fact_5222_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_pc(A,fun(B,A),Y)),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(set(B),nat,finite_card(B),A3)) ) ).

% prod_constant
tff(fact_5223_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_atLeastLessThan_int
tff(fact_5224_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_5225_card__insert__disjoint,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( finite_finite2(A,A3)
     => ( ~ aa(set(A),$o,member(A,Xb),A3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_5226_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_pd(A,fun(B,A),Y)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))),Y) ) ).

% sum_constant
tff(fact_5227_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( ~ aa(set(A),$o,member(A,A2),B4)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_5228_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or1337092689740270186AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_5229_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),K),zero_zero(code_integer)) = K ).

% minus_integer_code(1)
tff(fact_5230_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(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_5231_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(nat,B),Nat: nat] : aa(B,A,H,case_nat(B,F1,F22,Nat)) = case_nat(A,aa(B,A,H,F1),aa(fun(nat,B),fun(nat,A),aTP_Lamp_pe(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_5232_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_5233_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_5234_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_5235_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_5236_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X23: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X23)) = aa(nat,A,F22,X23) ).

% old.nat.simps(5)
tff(fact_5237_n__subsets,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( finite_finite2(A,A3)
     => ( aa(set(set(A)),nat,finite_card(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(nat,fun(set(A),$o),aTP_Lamp_pf(set(A),fun(nat,fun(set(A),$o)),A3),K))) = aa(nat,nat,binomial(aa(set(A),nat,finite_card(A),A3)),K) ) ) ).

% n_subsets
tff(fact_5238_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A)] : case_nat(A,F1,F22,zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_5239_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B4: set(A),A3: set(B),R: fun(B,fun(A,$o))] :
      ( finite_finite2(A,B4)
     => ( ! [A4: B] :
            ( aa(set(B),$o,member(B,A4),A3)
           => ? [B9: A] :
                ( aa(set(A),$o,member(A,B9),B4)
                & aa(A,$o,aa(B,fun(A,$o),R,A4),B9) ) )
       => ( ! [A12: B,A23: B,B3: A] :
              ( aa(set(B),$o,member(B,A12),A3)
             => ( aa(set(B),$o,member(B,A23),A3)
               => ( aa(set(A),$o,member(A,B3),B4)
                 => ( aa(A,$o,aa(B,fun(A,$o),R,A12),B3)
                   => ( aa(A,$o,aa(B,fun(A,$o),R,A23),B3)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_5240_card__insert__le,axiom,
    ! [A: $tType,A3: set(A),Xb: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),A3))) ).

% card_insert_le
tff(fact_5241_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_pg(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_5242_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_ph(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_5243_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S3: set(A),Ta: set(B),R2: fun(A,fun(B,$o)),K: fun(B,nat)] :
      ( finite_finite2(A,S3)
     => ( finite_finite2(B,Ta)
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),Ta)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_av(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S3),R2),X2))) = aa(B,nat,K,X2) ) )
         => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_pi(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Ta),R2)),S3) = aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,K),Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_5244_card__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( finite_finite2(A,A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_aj(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A3)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_5245_card__eq__sum,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_pj(A,nat)),A3) ).

% card_eq_sum
tff(fact_5246_card__2__iff_H,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X: A] :
          ( aa(set(A),$o,member(A,X),S)
          & ? [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),S)
              & ( X != Xa2 )
              & ! [Xb4: A] :
                  ( aa(set(A),$o,member(A,Xb4),S)
                 => ( ( Xb4 = X )
                    | ( Xb4 = Xa2 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_5247_card__ge__0__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
     => finite_finite2(A,A3) ) ).

% card_ge_0_finite
tff(fact_5248_card__insert__if,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),A3)) = $ite(aa(set(A),$o,member(A,Xb),A3),aa(set(A),nat,finite_card(A),A3),aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_insert_if
tff(fact_5249_card__Suc__eq__finite,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B5: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,B5),B10) )
          & ~ aa(set(A),$o,member(A,B5),B10)
          & ( aa(set(A),nat,finite_card(A),B10) = K )
          & finite_finite2(A,B10) ) ) ).

% card_Suc_eq_finite
tff(fact_5250_card__image__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( finite_finite2(A,A3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(A),nat,finite_card(A),A3)) ) ).

% card_image_le
tff(fact_5251_card__mono,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( finite_finite2(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ).

% card_mono
tff(fact_5252_card__seteq,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( finite_finite2(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B4)),aa(set(A),nat,finite_card(A),A3))
         => ( A3 = B4 ) ) ) ) ).

% card_seteq
tff(fact_5253_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C5: nat] :
      ( ! [G4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),G4),F4)
         => ( finite_finite2(A,G4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G4)),C5) ) )
     => ( finite_finite2(A,F4)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F4)),C5) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_5254_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Nb: nat,S: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),S))
     => ~ ! [T5: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T5),S)
           => ( ( aa(set(A),nat,finite_card(A),T5) = Nb )
             => ~ finite_finite2(A,T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_5255_card__1__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) )
     => ~ ! [X2: A] : A3 != aa(set(A),set(A),insert(A,X2),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_5256_card__less__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(A,B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3))) ) ) ) ).

% card_less_sym_Diff
tff(fact_5257_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% card_length
tff(fact_5258_card__le__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(A,B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3))) ) ) ) ).

% card_le_sym_Diff
tff(fact_5259_psubset__card__mono,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( finite_finite2(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ).

% psubset_card_mono
tff(fact_5260_card__less__Suc2,axiom,
    ! [M6: set(nat),I2: nat] :
      ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M6)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_pk(set(nat),fun(nat,fun(nat,$o)),M6),I2))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_pl(set(nat),fun(nat,fun(nat,$o)),M6),I2))) ) ) ).

% card_less_Suc2
tff(fact_5261_card__less__Suc,axiom,
    ! [M6: set(nat),I2: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M6)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_pk(set(nat),fun(nat,fun(nat,$o)),M6),I2)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_pl(set(nat),fun(nat,fun(nat,$o)),M6),I2))) ) ) ).

% card_less_Suc
tff(fact_5262_card__less,axiom,
    ! [M6: set(nat),I2: nat] :
      ( aa(set(nat),$o,member(nat,zero_zero(nat)),M6)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_pl(set(nat),fun(nat,fun(nat,$o)),M6),I2))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_5263_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_5264_sum__constant__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_pm(A,fun(B,A),Y)),A3) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,semiring_1_of_nat(real),aa(set(B),nat,finite_card(B),A3))),Y) ) ).

% sum_constant_scaleR
tff(fact_5265_sum__Suc,axiom,
    ! [A: $tType,F2: fun(A,nat),A3: set(A)] : aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_pn(fun(A,nat),fun(A,nat),F2)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(set(A),nat,finite_card(A),A3)) ).

% sum_Suc
tff(fact_5266_subset__card__intvl__is__intvl,axiom,
    ! [A3: set(nat),K: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A3),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3))))
     => ( A3 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_5267_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S: set(A),T2: set(B),R2: fun(A,fun(B,$o)),K: nat] :
      ( finite_finite2(A,S)
     => ( finite_finite2(B,T2)
       => ( ! [X2: B] :
              ( aa(set(B),$o,member(B,X2),T2)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_av(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S),R2),X2))) = K ) )
         => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_pi(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T2),R2)),S) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T2)) ) ) ) ) ).

% sum_multicount
tff(fact_5268_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Mb)),Nb)
    <=> case_nat($o,$false,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_5269_real__of__card,axiom,
    ! [A: $tType,A3: set(A)] : aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),A3)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_po(A,real)),A3) ).

% real_of_card
tff(fact_5270_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),Mb) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_pp(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc1
tff(fact_5271_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mb),aa(nat,nat,suc,Nb)) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_pq(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc2
tff(fact_5272_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K5: B] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),K5) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K5)) ) ) ).

% sum_bounded_above
tff(fact_5273_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),K5: B,F2: fun(A,B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K5),aa(A,B,F2,I3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)) ) ) ).

% sum_bounded_below
tff(fact_5274_card__gt__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
    <=> ( ( A3 != bot_bot(set(A)) )
        & finite_finite2(A,A3) ) ) ).

% card_gt_0_iff
tff(fact_5275_card__1__singleton__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X: A] : A3 = aa(set(A),set(A),insert(A,X),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_5276_card__eq__SucD,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
     => ? [B3: A,B8: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,B3),B8) )
          & ~ aa(set(A),$o,member(A,B3),B8)
          & ( aa(set(A),nat,finite_card(A),B8) = K )
          & ( ( K = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_5277_card__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B5: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,B5),B10) )
          & ~ aa(set(A),$o,member(A,B5),B10)
          & ( aa(set(A),nat,finite_card(A),B10) = K )
          & ( ( K = zero_zero(nat) )
           => ( B10 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_5278_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => ! [Xa2: A] :
                ( aa(set(A),$o,member(A,Xa2),A3)
               => ( X = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_5279_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B),F2: fun(A,B)] :
      ( finite_finite2(A,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),aa(set(A),set(B),image(A,B,F2),A3))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B4)),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% surj_card_le
tff(fact_5280_card__le__Suc__iff,axiom,
    ! [A: $tType,Nb: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),A3))
    <=> ? [A5: A,B10: set(A)] :
          ( ( A3 = aa(set(A),set(A),insert(A,A5),B10) )
          & ~ aa(set(A),$o,member(A,A5),B10)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),B10))
          & finite_finite2(A,B10) ) ) ).

% card_le_Suc_iff
tff(fact_5281_card__Diff1__le,axiom,
    ! [A: $tType,A3: set(A),Xb: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ).

% card_Diff1_le
tff(fact_5282_card__Diff__subset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( finite_finite2(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ).

% card_Diff_subset
tff(fact_5283_card__psubset,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( finite_finite2(A,B4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A3),B4) ) ) ) ).

% card_psubset
tff(fact_5284_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( finite_finite2(A,B4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))) ) ).

% diff_card_le_card_Diff
tff(fact_5285_card__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( finite_finite2(A,A3)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ak(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A3))),aa(nat,set(nat),set_ord_atMost(nat),Nb)) ) ) ).

% card_lists_length_le
tff(fact_5286_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M6: set(A)] :
      ( finite_finite2(A,M6)
     => ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M6)),M6) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_5287_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ao(nat,fun(A,$o),Nb)))),Nb) ) ) ).

% card_roots_unity
tff(fact_5288_card__le__Suc__Max,axiom,
    ! [S: set(nat)] :
      ( finite_finite2(nat,S)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S))) ) ).

% card_le_Suc_Max
tff(fact_5289_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N2)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_5290_card__sum__le__nat__sum,axiom,
    ! [S: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_df(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S)))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_df(nat,nat)),S)) ).

% card_sum_le_nat_sum
tff(fact_5291_card__nth__roots,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_lk(complex,fun(nat,fun(complex,$o)),C2),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_5292_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_da(nat,fun(complex,$o),Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_5293_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_df(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ).

% diff_Suc
tff(fact_5294_card__2__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X: A,Y3: A] :
          ( ( S = aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,Y3),bot_bot(set(A)))) )
          & ( X != Y3 ) ) ) ).

% card_2_iff
tff(fact_5295_card__3__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X: A,Y3: A,Z3: A] :
          ( ( S = aa(set(A),set(A),insert(A,X),aa(set(A),set(A),insert(A,Y3),aa(set(A),set(A),insert(A,Z3),bot_bot(set(A))))) )
          & ( X != Y3 )
          & ( Y3 != Z3 )
          & ( X != Z3 ) ) ) ).

% card_3_iff
tff(fact_5296_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),A3))
     => ( A3 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_5297_card__Suc__Diff1,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),$o,member(A,Xb),A3)
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_5298_card_Oinsert__remove,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_5299_card_Oremove,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),$o,member(A,Xb),A3)
       => ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_5300_card__Diff1__less__iff,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))
    <=> ( finite_finite2(A,A3)
        & aa(set(A),$o,member(A,Xb),A3) ) ) ).

% card_Diff1_less_iff
tff(fact_5301_card__Diff2__less,axiom,
    ! [A: $tType,A3: set(A),Xb: A,Y: A] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),$o,member(A,Xb),A3)
       => ( aa(set(A),$o,member(A,Y),A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_Diff2_less
tff(fact_5302_card__Diff1__less,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),$o,member(A,Xb),A3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Diff1_less
tff(fact_5303_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,W))),Nb)
        <=> case_nat($o,$false,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),Nb) ) ) ).

% bit_numeral_rec(1)
tff(fact_5304_card__Diff__singleton,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,Xb),A3)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_5305_card__Diff__singleton__if,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))) = $ite(aa(set(A),$o,member(A,Xb),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),one_one(nat)),aa(set(A),nat,finite_card(A),A3)) ).

% card_Diff_singleton_if
tff(fact_5306_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),Nb)
        <=> case_nat($o,$true,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),Nb) ) ) ).

% bit_numeral_rec(2)
tff(fact_5307_sum__norm__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S: set(A),F2: fun(A,B),K5: real] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),S)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),K5) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),S))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),S))),K5)) ) ) ).

% sum_norm_bound
tff(fact_5308_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F2: fun(A,B),Nb: B,K: nat] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I3))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),Nb) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),K)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),Nb)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Nb),K)) ) ) ) ) ).

% prod_le_power
tff(fact_5309_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A3: set(A),F2: fun(A,B),K5: B] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I3)),K5) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3))),K5)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_5310_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A3: set(A),F2: fun(A,B),K5: B] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),aa(B,B,aa(B,fun(B,B),divide_divide(B),K5),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A3)))) )
         => ( finite_finite2(A,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),K5) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_5311_card__insert__le__m1,axiom,
    ! [A: $tType,Nb: nat,Y: set(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),Y))),Nb) ) ) ).

% card_insert_le_m1
tff(fact_5312_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S: set(A),R2: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( finite_finite2(A,S)
         => ( finite_finite2(B,R2)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S)),R2)
             => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_pr(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)),S) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_ps(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G),F2)),R2) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_5313_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A2: A,B2: fun(A,B),C2: B] :
          ( finite_finite2(A,S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_pt(A,fun(fun(A,B),fun(B,fun(A,B))),A2),B2),C2)),S) = $ite(aa(set(A),$o,member(A,A2),S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),S)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C2),aa(set(A),nat,finite_card(A),S))) ) ) ) ).

% prod_gen_delta
tff(fact_5314_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ).

% polyfun_roots_card
tff(fact_5315_sum__le__card__Max,axiom,
    ! [A: $tType,A3: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image(A,nat,F2),A3)))) ) ).

% sum_le_card_Max
tff(fact_5316_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))),Nb) ) ) ) ) ).

% polyfun_rootbound
tff(fact_5317_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xb: A,F2: fun(nat,A),Nb: nat] :
      case_nat(A,Xb,F2,Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_5318_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= 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),aa(num,num,bit0,one2))),code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),
            $ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_5319_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( finite_finite2(A,A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A3))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_pu(set(A),fun(nat,fun(list(A),$o)),A3),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_df(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_5320_distinct__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( distinct(A,list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I2)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_5321_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( finite_finite2(A,A3)
     => finite_finite2(list(A),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_pu(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_5322_finite__distinct__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A3 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_5323_plus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xaa),Xb)) ).

% plus_integer.abs_eq
tff(fact_5324_times__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),times_times(int),Xaa),Xb)) ).

% times_integer.abs_eq
tff(fact_5325_minus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),minus_minus(int),Xaa),Xb)) ).

% minus_integer.abs_eq
tff(fact_5326_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( I4 != J3 )
               => ( aa(nat,A,nth(A,Xs),I4) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_5327_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,J: nat] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs),J) )
          <=> ( I2 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_5328_card__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
     => distinct(A,Xs) ) ).

% card_distinct
tff(fact_5329_distinct__card,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ).

% distinct_card
tff(fact_5330_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
       => ? [X2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X2) = Xb )
            & ! [Y5: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y5),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y5) = Xb ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_5331_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat),B4: set(A)] :
      ( distinct(A,Xs)
     => ( ( A3 = aa(nat,set(nat),set_ord_lessThan(nat),aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B4 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A3,B4) ) ) ) ).

% bij_betw_nth
tff(fact_5332_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I2: nat] :
      ( distinct(A,Xs)
     => ( ~ aa(set(A),$o,member(A,A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,aa(nat,A,nth(A,Xs),I2)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs,I2,A2)) ) ) ).

% distinct_list_update
tff(fact_5333_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xb)) = aa(set(A),set(A),insert(A,Xb),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,aa(nat,A,nth(A,Xs),Nb)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_5334_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(A),nat,finite_card(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_pv(nat,fun(set(A),fun(list(A),$o)),K),A3))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_df(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_5335_integer__of__num_I3_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa(num,num,bit1,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        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),plus_plus(code_integer),k),k)),one_one(code_integer)) ) ).

% integer_of_num(3)
tff(fact_5336_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_pw(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ) ).

% int_of_integer_code
tff(fact_5337_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_px(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% num_of_integer_code
tff(fact_5338_plus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% plus_integer.rep_eq
tff(fact_5339_times__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),times_times(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% times_integer.rep_eq
tff(fact_5340_minus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% minus_integer.rep_eq
tff(fact_5341_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_5342_integer__of__num_I2_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa(num,num,bit0,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k) ) ).

% integer_of_num(2)
tff(fact_5343_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(aa(num,num,bit0,one2)) = aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)) ).

% integer_of_num_triv(2)
tff(fact_5344_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_py(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% nat_of_integer_code
tff(fact_5345_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o),F1: A,F22: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P,case_nat(A,F1,F22,Nat))
    <=> ( ( ( Nat = zero_zero(nat) )
         => aa(A,$o,P,F1) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => aa(A,$o,P,aa(nat,A,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_5346_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),F1: A,F22: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P,case_nat(A,F1,F22,Nat))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ aa(A,$o,P,F1) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ aa(A,$o,P,aa(nat,A,F22,pred(Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_5347_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_of_integer_code_post(3)
tff(fact_5348_nat__of__integer__code__post_I2_J,axiom,
    code_nat_of_integer(one_one(code_integer)) = one_one(nat) ).

% nat_of_integer_code_post(2)
tff(fact_5349_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_df(nat,nat),Nat) ).

% pred_def
tff(fact_5350_card__Pow,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow2(A,A3)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Pow
tff(fact_5351_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,fun(A,A)),V: num,Nb: nat] :
      aa(nat,A,rec_nat(A,A2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = $let(
        pv: nat,
        pv:= pred_numeral(V),
        aa(A,A,aa(nat,fun(A,A),F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb)),aa(nat,A,rec_nat(A,A2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb))) ) ).

% rec_nat_add_eq_if
tff(fact_5352_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_pz(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_5353_old_Onat_Osimps_I7_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,fun(A,A)),Nat: nat] : aa(nat,A,rec_nat(A,F1,F22),aa(nat,nat,suc,Nat)) = aa(A,A,aa(nat,fun(A,A),F22,Nat),aa(nat,A,rec_nat(A,F1,F22),Nat)) ).

% old.nat.simps(7)
tff(fact_5354_old_Onat_Osimps_I6_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,fun(A,A))] : aa(nat,A,rec_nat(A,F1,F22),zero_zero(nat)) = F1 ).

% old.nat.simps(6)
tff(fact_5355_rec__nat__numeral,axiom,
    ! [A: $tType,A2: A,F2: fun(nat,fun(A,A)),V: num] :
      aa(nat,A,rec_nat(A,A2,F2),aa(num,nat,numeral_numeral(nat),V)) = $let(
        pv: nat,
        pv:= pred_numeral(V),
        aa(A,A,aa(nat,fun(A,A),F2,pv),aa(nat,A,rec_nat(A,A2,F2),pv)) ) ).

% rec_nat_numeral
tff(fact_5356_Pow__def,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,A3) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_an(set(A),fun(set(A),$o),A3)) ).

% Pow_def
tff(fact_5357_binomial__def,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(Nb),K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),$o),aTP_Lamp_qa(nat,fun(nat,fun(set(nat),$o)),Nb),K))) ).

% binomial_def
tff(fact_5358_subseqs__powset,axiom,
    ! [A: $tType,Xs: list(A)] : 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))) = pow2(A,aa(list(A),set(A),set2(A),Xs)) ).

% subseqs_powset
tff(fact_5359_old_Orec__nat__def,axiom,
    ! [A: $tType,X4: A,Xa: fun(nat,fun(A,A)),Xb2: nat] : aa(nat,A,rec_nat(A,X4,Xa),Xb2) = the(A,rec_set_nat(A,X4,Xa,Xb2)) ).

% old.rec_nat_def
tff(fact_5360_rec__nat__0__imp,axiom,
    ! [A: $tType,F2: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F2 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F2,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_5361_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F2: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),Nb: nat] :
      ( ( F2 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F2,aa(nat,nat,suc,Nb)) = aa(A,A,aa(nat,fun(A,A),F22,Nb),aa(nat,A,F2,Nb)) ) ) ).

% rec_nat_Suc_imp
tff(fact_5362_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_5363_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xb),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xb)))) ) ) ).

% prod_decode_aux.elims
tff(fact_5364_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Mb: nat] :
      nat_prod_decode_aux(K,Mb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),K),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),Mb)),nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,K)))) ).

% prod_decode_aux.simps
tff(fact_5365_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se4197421643247451524op_bit(A,Nb,zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_5366_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : bit_se4197421643247451524op_bit(A,Mb,bit_se4197421643247451524op_bit(A,Nb,A2)) = bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb),A2) ) ).

% drop_bit_drop_bit
tff(fact_5367_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,Nb,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,Nb,A2)),bit_se4197421643247451524op_bit(A,Nb,B2)) ) ).

% drop_bit_and
tff(fact_5368_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,Nb,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,Nb,A2)),bit_se4197421643247451524op_bit(A,Nb,B2)) ) ).

% drop_bit_or
tff(fact_5369_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : bit_se4197421643247451524op_bit(A,Nb,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,Nb,A2)),bit_se4197421643247451524op_bit(A,Nb,B2)) ) ).

% drop_bit_xor
tff(fact_5370_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,B2: $o] :
          bit_se4197421643247451524op_bit(A,Nb,aa($o,A,zero_neq_one_of_bool(A),(B2))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( Nb = zero_zero(nat) )
            & (B2) )) ) ).

% drop_bit_of_bool
tff(fact_5371_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se4197421643247451524op_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_5372_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se4197421643247451524op_bit(int,Nb,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_5373_drop__bit__minus__one,axiom,
    ! [Nb: nat] : bit_se4197421643247451524op_bit(int,Nb,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_5374_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = bit_se4197421643247451524op_bit(A,Nb,aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_5375_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = bit_se4197421643247451524op_bit(A,Nb,aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_5376_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se4197421643247451524op_bit(A,Nb,one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_5377_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),aa(num,num,bit0,K))) = bit_se4197421643247451524op_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit0
tff(fact_5378_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_5379_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = bit_se4197421643247451524op_bit(int,Nb,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_5380_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),aa(num,num,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_5381_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = bit_se4197421643247451524op_bit(int,Nb,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_5382_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se4197421643247451524op_bit(nat,Mb,Nb)) = bit_se4197421643247451524op_bit(A,Mb,aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_drop_bit
tff(fact_5383_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,Mb: nat] : bit_se4197421643247451524op_bit(A,Nb,aa(nat,A,semiring_1_of_nat(A),Mb)) = aa(nat,A,semiring_1_of_nat(A),bit_se4197421643247451524op_bit(nat,Nb,Mb)) ) ).

% drop_bit_of_nat
tff(fact_5384_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = A2 )
        <=> ( bit_se4197421643247451524op_bit(A,Nb,A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_5385_drop__bit__push__bit__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] : bit_se4197421643247451524op_bit(int,Mb,bit_se4730199178511100633sh_bit(int,Nb,K)) = bit_se4197421643247451524op_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),bit_se4730199178511100633sh_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb),K)) ).

% drop_bit_push_bit_int
tff(fact_5386_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),bit_se4197421643247451524op_bit(A,Nb,A2)) = bit_se4197421643247451524op_bit(A,Nb,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),A2)) ) ).

% take_bit_drop_bit
tff(fact_5387_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : bit_se4197421643247451524op_bit(A,Mb,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)),bit_se4197421643247451524op_bit(A,Mb,A2)) ) ).

% drop_bit_take_bit
tff(fact_5388_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),bit_se4730199178511100633sh_bit(A,Nb,one_one(A))) = bit_se4197421643247451524op_bit(A,Nb,A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_5389_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se4197421643247451524op_bit(A,Nb,A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_5390_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se4730199178511100633sh_bit(A,Nb,bit_se4197421643247451524op_bit(A,Nb,A2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = A2 ) ).

% bits_ident
tff(fact_5391_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4197421643247451524op_bit(A,Nb,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),bit_se4197421643247451524op_bit(A,Nb,A2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% drop_bit_half
tff(fact_5392_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( bit_se4197421643247451524op_bit(A,Nb,A2) = A2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_5393_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : bit_se4197421643247451524op_bit(int,Nb,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),aa(num,num,bit0,one2))),Nb)) ).

% drop_bit_int_def
tff(fact_5394_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb),A2) = bit_se4197421643247451524op_bit(A,Nb,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% drop_bit_Suc
tff(fact_5395_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se4197421643247451524op_bit(A,Nb,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),aa(num,num,bit0,one2))),Nb)) ) ).

% drop_bit_eq_div
tff(fact_5396_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se4197421643247451524op_bit(A,Nb,A2)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_5397_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se4197421643247451524op_bit(A,Nb,A2))
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_5398_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,Mb: nat,A2: A] : bit_se4730199178511100633sh_bit(A,Nb,aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),bit_se4197421643247451524op_bit(A,Nb,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),Mb),Nb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)))) ) ).

% slice_eq_mask
tff(fact_5399_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          bit_se4197421643247451524op_bit(A,Nb,A2) = $ite(Nb = zero_zero(nat),A2,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% drop_bit_rec
tff(fact_5400_Suc__0__mod__numeral,axiom,
    ! [K: num] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_mod_numeral
tff(fact_5401_Suc__0__div__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_div_numeral
tff(fact_5402_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,$o)),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o)),aTP_Lamp_qb(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% bit_cut_integer_code
tff(fact_5403_prod_Ocollapse,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) = Prod ).

% prod.collapse
tff(fact_5404_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_div_numeral
tff(fact_5405_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),K),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_mod_numeral
tff(fact_5406_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : bit_se4197421643247451524op_bit(nat,Nb,aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),Nb = zero_zero(nat)) ).

% drop_bit_of_Suc_0
tff(fact_5407_fst__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(Mb,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),Nb) ).

% fst_divmod_nat
tff(fact_5408_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,Nb)) ) ).

% one_div_numeral
tff(fact_5409_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),Nb)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,Nb)) ) ).

% one_mod_numeral
tff(fact_5410_snd__conv,axiom,
    ! [B: $tType,A: $tType,X1: B,X23: A] : aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X1),X23)) = X23 ).

% snd_conv
tff(fact_5411_snd__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: B,Y: A,A2: A] :
      ( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_5412_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X23: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X23)) = X1 ).

% fst_conv
tff(fact_5413_fst__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,A2: A] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y)) = A2 )
     => ( Xb = A2 ) ) ).

% fst_eqD
tff(fact_5414_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,Ta: product_prod(A,B)] : Ta = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Ta)),aa(product_prod(A,B),B,product_snd(A,B),Ta)) ).

% surjective_pairing
tff(fact_5415_prod_Oexhaust__sel,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).

% prod.exhaust_sel
tff(fact_5416_prod_Osplit__sel,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F2: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),Prod))
    <=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
       => aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F2,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel
tff(fact_5417_prod_Osplit__sel__asm,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F2: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F2),Prod))
    <=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
          & ~ aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F2,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_5418_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : fChoice(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)) = fChoice(product_prod(A,B),aTP_Lamp_qc(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% Eps_case_prod
tff(fact_5419_fst__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,aa(fun(A,fun(B,A)),fun(product_prod(A,B),A),product_case_prod(A,B,A),aTP_Lamp_mw(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_5420_snd__def,axiom,
    ! [A: $tType,B: $tType,Prod: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Prod) = aa(product_prod(B,A),A,aa(fun(B,fun(A,A)),fun(product_prod(B,A),A),product_case_prod(B,A,A),aTP_Lamp_qd(B,fun(A,A))),Prod) ).

% snd_def
tff(fact_5421_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X4: fun(A,fun(B,C)),Xa: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X4),Xa) = aa(B,C,aa(A,fun(B,C),X4,aa(product_prod(A,B),A,product_fst(A,B),Xa)),aa(product_prod(A,B),B,product_snd(A,B),Xa)) ).

% case_prod_unfold
tff(fact_5422_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,fun(B,C)),X4: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),X4) = aa(B,C,aa(A,fun(B,C),F2,aa(product_prod(A,B),A,product_fst(A,B),X4)),aa(product_prod(A,B),B,product_snd(A,B),X4)) ).

% case_prod_beta'
tff(fact_5423_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,F2: fun(D,fun(B,C)),G: fun(A,D)] : aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_qe(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),F2),G) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_qf(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),F2),G)) ).

% split_comp_eq
tff(fact_5424_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)) = the(product_prod(A,B),aTP_Lamp_qc(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% The_case_prod
tff(fact_5425_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Xb: A,Y: B,A2: product_prod(A,B)] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Xb),Y)
     => ( ( A2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y) )
       => aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),A2)),aa(product_prod(A,B),B,product_snd(A,B),A2)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_5426_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : bit_se4197421643247451524op_bit(nat,Nb,aa(int,nat,nat2,K)) = aa(int,nat,nat2,bit_se4197421643247451524op_bit(int,Nb,K)) ).

% drop_bit_nat_eq
tff(fact_5427_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% fst_divmod
tff(fact_5428_snd__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,Mb,Nb)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Nb)) ) ).

% snd_divmod
tff(fact_5429_drop__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : bit_se4197421643247451524op_bit(nat,Nb,Mb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)) ).

% drop_bit_nat_def
tff(fact_5430_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),dvd_dvd(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),K)) ).

% bit_cut_integer_def
tff(fact_5431_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        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),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(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_qg(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_zero(code_integer),
            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),zero_zero(code_integer)),K),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),
              $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(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_qh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_5432_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),Nb: nat,Xs: list(A)] :
      ( aa(set(product_prod(nat,A)),$o,member(product_prod(nat,A),P2),aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,Nb,Xs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(product_prod(nat,A),nat,product_fst(nat,A),P2))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),Nb)) = aa(product_prod(nat,A),A,product_snd(nat,A),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_5433_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),P2: product_prod(B,A),Q: fun(C,$o),F2: fun(B,fun(A,C))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2))
     => ( ! [X2: B,Y4: A] :
            ( aa(B,$o,aa(A,fun(B,$o),P,Y4),X2)
           => aa(C,$o,Q,aa(A,C,aa(B,fun(A,C),F2,X2),Y4)) )
       => aa(C,$o,Q,aa(product_prod(B,A),C,aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),F2),P2)) ) ) ).

% exE_realizer
tff(fact_5434_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,B),Xb: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Xb),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),aa(C,B,F2,Y)) ).

% apsnd_conv
tff(fact_5435_length__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,Nb,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_enumerate
tff(fact_5436_nth__enumerate__eq,axiom,
    ! [A: $tType,Mb: nat,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,Nb,Xs)),Mb) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb)),aa(nat,A,nth(A,Xs),Mb)) ) ) ).

% nth_enumerate_eq
tff(fact_5437_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Y: A,Xb: B] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Y),Xb)
     => aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y))) ) ).

% exI_realizer
tff(fact_5438_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),P2: A,Q: fun(B,$o),Q2: B] :
      ( aa(A,$o,P,P2)
     => ( aa(B,$o,Q,Q2)
       => ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q2)))
          & aa(B,$o,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_5439_minus__one__mod__numeral,axiom,
    ! [Nb: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),Nb)) = adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Nb))) ).

% minus_one_mod_numeral
tff(fact_5440_one__mod__minus__numeral,axiom,
    ! [Nb: num] : modulo_modulo(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Nb)))) ).

% one_mod_minus_numeral
tff(fact_5441_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Y)))))) ).

% bezw.simps
tff(fact_5442_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R: int] :
      adjust_mod(L,R) = $ite(R = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R)) ).

% Divides.adjust_mod_def
tff(fact_5443_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Y)
     => ( bezw(Xb,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,Xb,Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Y))))) ) ) ).

% bezw_non_0
tff(fact_5444_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_5445_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xb,Xaa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Xaa)))))) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_5446_normalize__def,axiom,
    ! [P2: product_prod(int,int)] :
      normalize(P2) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
        $let(
          a2: int,
          a2:= aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
          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(product_prod(int,int),int,product_fst(int,int),P2)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a2)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(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)),
          $let(
            a2: int,
            a2:= aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))),
            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(product_prod(int,int),int,product_fst(int,int),P2)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a2)) ) ) ) ).

% normalize_def
tff(fact_5447_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,nat),G: fun(B,nat),P2: product_prod(A,B)] : basic_BNF_size_prod(A,B,F2,G,P2) = 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,F2,aa(product_prod(A,B),A,product_fst(A,B),P2))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P2)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_5448_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),one_one(A)) = one_one(A) ) ).

% gcd.bottom_right_bottom
tff(fact_5449_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),one_one(A)),A2) = one_one(A) ) ).

% gcd.bottom_left_bottom
tff(fact_5450_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_add2
tff(fact_5451_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Mb),Nb)),Nb) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_add1
tff(fact_5452_gcd__exp,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A2: A,Nb: nat,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),Nb) ) ).

% gcd_exp
tff(fact_5453_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,Nb: num] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(num,A,numeral_numeral(A),Nb)) ) ).

% gcd_neg_numeral_2
tff(fact_5454_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: num,A2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),A2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(num,A,numeral_numeral(A),Nb)),A2) ) ).

% gcd_neg_numeral_1
tff(fact_5455_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),one_one(A))
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_5456_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,K: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),Mb)),Nb)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_add_mult
tff(fact_5457_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: A,Mb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),Mb)),Nb) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_diff2
tff(fact_5458_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Mb: A,Nb: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Mb),Nb)),Nb) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),Mb),Nb) ) ).

% gcd_diff1
tff(fact_5459_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,K: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2)) ) ).

% gcd_dvd_prod
tff(fact_5460_bezout__int,axiom,
    ! [Xb: int,Y: int] :
    ? [U2: int,V2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U2),Xb)),aa(int,int,aa(int,fun(int,int),times_times(int),V2),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xb),Y) ).

% bezout_int
tff(fact_5461_gcd__mult__distrib__int,axiom,
    ! [K: int,Mb: int,Nb: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Mb),Nb)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),Mb)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Nb)) ).

% gcd_mult_distrib_int
tff(fact_5462_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_5463_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_5464_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit2
tff(fact_5465_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit1
tff(fact_5466_gcd__is__Max__divisors__int,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Mb),Nb) = aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_qi(int,fun(int,fun(int,$o)),Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_5467_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun(nat,$o),Mb: nat] :
      ( ! [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2)
         => aa(nat,$o,P,K2) )
     => ( ! [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => ( ! [I: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),I)
                 => aa(nat,$o,P,I) )
             => aa(nat,$o,P,K2) ) )
       => aa(nat,$o,P,Mb) ) ) ).

% nat_descend_induct
tff(fact_5468_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q2: product_prod(A,B),F2: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P2: product_prod(A,B)] :
      ( ! [X2: A,Y4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y4) = Q2 )
         => ( aa(B,C,aa(A,fun(B,C),F2,X2),Y4) = aa(B,C,aa(A,fun(B,C),G,X2),Y4) ) )
     => ( ( P2 = Q2 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2),P2) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q2) ) ) ) ).

% split_cong
tff(fact_5469_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Y )
     => ( accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xb),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xb)))) )
           => ~ accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_5470_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,Xb,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_5471_nth__rotate1,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,rotate1(A,Xs)),Nb) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,Nb),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_5472_gcd__1__nat,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_5473_gcd__Suc__0,axiom,
    ! [Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_5474_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb))
    <=> ( ( Mb != zero_zero(nat) )
        | ( Nb != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_5475_in__set__remove1,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( ( A2 != B2 )
     => ( aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),remove1(A,B2,Xs)))
      <=> aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% in_set_remove1
tff(fact_5476_set__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),rotate1(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rotate1
tff(fact_5477_length__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),rotate1(A,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate1
tff(fact_5478_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5479_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2) ) ).

% gcd_le1_nat
tff(fact_5480_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2) ) ).

% gcd_le2_nat
tff(fact_5481_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_5482_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_5483_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_5484_notin__set__remove1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] :
      ( ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),remove1(A,Y,Xs))) ) ).

% notin_set_remove1
tff(fact_5485_remove1__idem,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ( remove1(A,Xb,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_5486_set__remove1__subset,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,Xb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_remove1_subset
tff(fact_5487_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X2: nat,Y4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) ) ).

% bezout_nat
tff(fact_5488_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X2: nat,Y4: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_5489_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Mb),Nb) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_qj(nat,fun(nat,fun(nat,$o)),Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_5490_length__remove1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      aa(list(A),nat,size_size(list(A)),remove1(A,Xb,Xs)) = $ite(aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_remove1
tff(fact_5491_bezw__aux,axiom,
    ! [Xb: nat,Y: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xb,Y))),aa(nat,int,semiring_1_of_nat(int),Xb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xb,Y))),aa(nat,int,semiring_1_of_nat(int),Y))) ).

% bezw_aux
tff(fact_5492_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xb),Xaa) = Y )
     => ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),Xb,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),modulo_modulo(nat,Xb,Xaa))) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_5493_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or5935395276787703475ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_5494_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set(nat)] :
      ( ~ finite_finite2(nat,S)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N4)
          & aa(set(nat),$o,member(nat,N4),S) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_5495_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I2),set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I2),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_5496_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_5497_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_5498_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A2,B2) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_5499_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioo_iff
tff(fact_5500_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A2,B2)) ) ) ).

% infinite_Ioo
tff(fact_5501_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_5502_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_5503_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_5504_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_5505_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_5506_unbounded__k__infinite,axiom,
    ! [K: nat,S: set(nat)] :
      ( ! [M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),M)
         => ? [N8: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N8)
              & aa(set(nat),$o,member(nat,N8),S) ) )
     => ~ finite_finite2(nat,S) ) ).

% unbounded_k_infinite
tff(fact_5507_infinite__nat__iff__unbounded,axiom,
    ! [S: set(nat)] :
      ( ~ finite_finite2(nat,S)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N4)
          & aa(set(nat),$o,member(nat,N4),S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_5508_finite__enumerate,axiom,
    ! [S: set(nat)] :
      ( finite_finite2(nat,S)
     => ? [R3: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R3,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(nat),nat,finite_card(nat),S)))
          & ! [N8: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N8),aa(set(nat),nat,finite_card(nat),S))
             => aa(set(nat),$o,member(nat,aa(nat,nat,R3,N8)),S) ) ) ) ).

% finite_enumerate
tff(fact_5509_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),K) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),neg_numeral_sub(int,Nb,one2)),K)) ).

% xor_minus_numerals(1)
tff(fact_5510_xor__minus__numerals_I2_J,axiom,
    ! [K: int,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),neg_numeral_sub(int,Nb,one2))) ).

% xor_minus_numerals(2)
tff(fact_5511_sub__num__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_sub(A,one2,one2) = zero_zero(A) ) ) ).

% sub_num_simps(1)
tff(fact_5512_diff__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,Mb,Nb) ) ).

% diff_numeral_simps(1)
tff(fact_5513_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),aa(nat,nat,suc,L)) ).

% card_greaterThanLessThan
tff(fact_5514_sub__num__simps_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit0,K),aa(num,num,bit0,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(6)
tff(fact_5515_sub__num__simps_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K),aa(num,num,bit1,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(9)
tff(fact_5516_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Mb,Nb) ) ).

% add_neg_numeral_simps(1)
tff(fact_5517_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,Nb,Mb) ) ).

% add_neg_numeral_simps(2)
tff(fact_5518_semiring__norm_I166_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(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),neg_numeral_sub(A,V,W)),Y) ) ).

% semiring_norm(166)
tff(fact_5519_semiring__norm_I167_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(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W,V)),Y) ) ).

% semiring_norm(167)
tff(fact_5520_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Nb,Mb) ) ).

% diff_numeral_simps(4)
tff(fact_5521_sub__num__simps_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit0,K),aa(num,num,bit1,L)) = neg_numeral_dbl_dec(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(7)
tff(fact_5522_sub__num__simps_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K),aa(num,num,bit0,L)) = neg_numeral_dbl_inc(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(8)
tff(fact_5523_diff__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,one2,Nb) ) ).

% diff_numeral_special(1)
tff(fact_5524_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),Mb)),one_one(A)) = neg_numeral_sub(A,Mb,one2) ) ).

% diff_numeral_special(2)
tff(fact_5525_sub__num__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_sub(A,aa(num,num,bit1,K),one2) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,K)) ) ).

% sub_num_simps(5)
tff(fact_5526_not__minus__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Nb,one2) ) ).

% not_minus_numeral_eq
tff(fact_5527_sub__num__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_sub(A,aa(num,num,bit0,K),one2) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% sub_num_simps(4)
tff(fact_5528_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))) = neg_numeral_sub(A,one2,Mb) ) ).

% add_neg_numeral_special(1)
tff(fact_5529_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),one_one(A)) = neg_numeral_sub(A,one2,Mb) ) ).

% add_neg_numeral_special(2)
tff(fact_5530_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Mb)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,Mb,one2) ) ).

% add_neg_numeral_special(3)
tff(fact_5531_add__neg__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Nb)) = neg_numeral_sub(A,Nb,one2) ) ).

% add_neg_numeral_special(4)
tff(fact_5532_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,Mb,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb)) ) ).

% minus_sub_one_diff_one
tff(fact_5533_diff__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = neg_numeral_sub(A,Nb,one2) ) ).

% diff_numeral_special(7)
tff(fact_5534_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Mb: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Mb))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,Mb) ) ).

% diff_numeral_special(8)
tff(fact_5535_sub__num__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,aa(num,num,bit1,L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,L))) ) ).

% sub_num_simps(3)
tff(fact_5536_sub__num__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,aa(num,num,bit0,L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_5537_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F2: fun(A,B),A3: set(A),Xb: A,Y: A] :
          ( strict_mono_on(A,B,F2,A3)
         => ( aa(set(A),$o,member(A,Xb),A3)
           => ( aa(set(A),$o,member(A,Y),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_5538_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A),R: A,S3: A] :
          ( strict_mono_on(A,B,F2,A3)
         => ( aa(set(A),$o,member(A,R),A3)
           => ( aa(set(A),$o,member(A,S3),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R),S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R)),aa(A,B,F2,S3)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_5539_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [R3: A,S2: A] :
              ( aa(set(A),$o,member(A,R3),A3)
             => ( aa(set(A),$o,member(A,S2),A3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R3),S2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R3)),aa(A,B,F2,S2)) ) ) )
         => strict_mono_on(A,B,F2,A3) ) ) ).

% strict_mono_onI
tff(fact_5540_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( strict_mono_on(A,B,F2,A3)
        <=> ! [R5: A,S7: A] :
              ( ( aa(set(A),$o,member(A,R5),A3)
                & aa(set(A),$o,member(A,S7),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),R5),S7) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R5)),aa(A,B,F2,S7)) ) ) ) ).

% strict_mono_on_def
tff(fact_5541_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_5542_neg__numeral__class_Osub__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,K,L) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) ) ).

% neg_numeral_class.sub_def
tff(fact_5543_tanh__real__bounds,axiom,
    ! [Xb: real] : aa(set(real),$o,member(real,aa(real,real,tanh(real),Xb)),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) ).

% tanh_real_bounds
tff(fact_5544_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,Nb,Mb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Mb),Nb) ) ) ).

% sub_non_negative
tff(fact_5545_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),neg_numeral_sub(A,Nb,Mb)),zero_zero(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),Mb) ) ) ).

% sub_non_positive
tff(fact_5546_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),neg_numeral_sub(A,Nb,Mb)),zero_zero(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Nb),Mb) ) ) ).

% sub_negative
tff(fact_5547_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Mb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),neg_numeral_sub(A,Nb,Mb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Mb),Nb) ) ) ).

% sub_positive
tff(fact_5548_sub__inc__One__eq,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : neg_numeral_sub(A,inc(Nb),one2) = aa(num,A,numeral_numeral(A),Nb) ) ).

% sub_inc_One_eq
tff(fact_5549_minus__numeral__eq__not__sub__one,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,bit_ri4277139882892585799ns_not(A),neg_numeral_sub(A,Nb,one2)) ) ).

% minus_numeral_eq_not_sub_one
tff(fact_5550_sub__BitM__One__eq,axiom,
    ! [Nb: num] : neg_numeral_sub(int,bitM(Nb),one2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),neg_numeral_sub(int,Nb,one2)) ).

% sub_BitM_One_eq
tff(fact_5551_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xb: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xb,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_5552_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xb: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xb,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_5553_bounded__linear__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V4916620083959148203axioms(A,B,F2)
        <=> ? [K6: real] :
            ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K6)) ) ) ).

% bounded_linear_axioms_def
tff(fact_5554_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_5555_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_5556_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),A2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_5557_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_5558_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,C2: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),C2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C2)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_5559_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,A2: B,B2: B,C2: B] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),A2)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),A2),aa(B,B,aa(B,fun(B,B),minus_minus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C2)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_5560_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,Nb: nat] :
          ( nO_MATCH(A,A,one_one(A),Xb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),Nb) = 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))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) ) ) ) ).

% power_minus'
tff(fact_5561_scale__left__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,C2: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_5562_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,C2: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_5563_bounded__linear__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( ? [K8: real] :
            ! [X2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K8))
         => real_V4916620083959148203axioms(A,B,F2) ) ) ).

% bounded_linear_axioms.intro
tff(fact_5564_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(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_qk(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_funpow
tff(fact_5565_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F2: fun(C,A),G: fun(D,B),Xb: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G),Xb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,aa(product_prod(C,D),C,product_fst(C,D),Xb))),aa(D,B,G,aa(product_prod(C,D),D,product_snd(C,D),Xb))) ).

% apfst_apsnd
tff(fact_5566_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F2: fun(C,B),G: fun(D,A),Xb: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F2),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),Xb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G,aa(product_prod(D,C),D,product_fst(D,C),Xb))),aa(C,B,F2,aa(product_prod(D,C),C,product_snd(D,C),Xb))) ).

% apsnd_apfst
tff(fact_5567_Suc__funpow,axiom,
    ! [Nb: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),Nb),suc) = aa(nat,fun(nat,nat),plus_plus(nat),Nb) ).

% Suc_funpow
tff(fact_5568_funpow__0,axiom,
    ! [A: $tType,F2: fun(A,A),Xb: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F2),Xb) = Xb ).

% funpow_0
tff(fact_5569_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F2: fun(C,A),Xb: C,Y: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Xb),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,Xb)),Y) ).

% apfst_conv
tff(fact_5570_funpow__mult,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Nb)),F2) ).

% funpow_mult
tff(fact_5571_funpow__swap1,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat,Xb: A] : aa(A,A,F2,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),Xb)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),aa(A,A,F2,Xb)) ).

% funpow_swap1
tff(fact_5572_funpow__mod__eq,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A),Xb: A,Mb: nat] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),Xb) = Xb )
     => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),modulo_modulo(nat,Mb,Nb)),F2),Xb) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2),Xb) ) ) ).

% funpow_mod_eq
tff(fact_5573_funpow__times__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [F2: fun(A,nat),Xb: A] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(A,nat,F2,Xb)),aa(A,fun(A,A),times_times(A),Xb)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(A,nat,F2,Xb))) ) ).

% funpow_times_power
tff(fact_5574_bij__betw__funpow,axiom,
    ! [A: $tType,F2: fun(A,A),S: set(A),Nb: nat] :
      ( bij_betw(A,A,F2,S,S)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),S,S) ) ).

% bij_betw_funpow
tff(fact_5575_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A2) ) ).

% numeral_add_unfold_funpow
tff(fact_5576_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),Nb) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_5577_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),K) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% numeral_unfold_funpow
tff(fact_5578_relpowp__fun__conv,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Y)
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = Xb )
          & ( aa(nat,A,F5,Nb) = Y )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,F5,I4)),aa(nat,A,F5,aa(nat,nat,suc,I4))) ) ) ) ).

% relpowp_fun_conv
tff(fact_5579_relpowp__1,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),one_one(nat)),P) = P ).

% relpowp_1
tff(fact_5580_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_5581_relpowp__Suc__E,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z)
     => ~ ! [Y4: A] :
            ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Y4)
           => ~ aa(A,$o,aa(A,fun(A,$o),P,Y4),Z) ) ) ).

% relpowp_Suc_E
tff(fact_5582_relpowp__Suc__I,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Y: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Y)
     => ( aa(A,$o,aa(A,fun(A,$o),P,Y),Z)
       => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z) ) ) ).

% relpowp_Suc_I
tff(fact_5583_relpowp__Suc__D2,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z)
     => ? [Y4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y4)
          & aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Y4),Z) ) ) ).

% relpowp_Suc_D2
tff(fact_5584_relpowp__Suc__E2,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z)
     => ~ ! [Y4: A] :
            ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y4)
           => ~ aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Y4),Z) ) ) ).

% relpowp_Suc_E2
tff(fact_5585_relpowp__Suc__I2,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A,Y: A,Nb: nat,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y)
     => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Y),Z)
       => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z) ) ) ).

% relpowp_Suc_I2
tff(fact_5586_relpowp__E2,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y4: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y4)
               => ~ aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M),P),Y4),Z) ) ) ) ) ).

% relpowp_E2
tff(fact_5587_relpowp__E,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(A,$o)),Xb: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),P),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y4: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M),P),Xb),Y4)
               => ~ aa(A,$o,aa(A,fun(A,$o),P,Y4),Z) ) ) ) ) ).

% relpowp_E
tff(fact_5588_relpowp__bot,axiom,
    ! [A: $tType,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),bot_bot(fun(A,fun(A,$o)))) = bot_bot(fun(A,fun(A,$o))) ) ) ).

% relpowp_bot
tff(fact_5589_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q2: product_prod(A,B),F2: fun(C,A),P2: product_prod(C,B)] :
      ( ( Q2 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F2),P2) )
     => ~ ! [X2: C,Y4: B] :
            ( ( P2 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X2),Y4) )
           => ( Q2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F2,X2)),Y4) ) ) ) ).

% apfst_convE
tff(fact_5590_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_al(nat,fun(nat,$o)),aTP_Lamp_am(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_5591_set__removeAll,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),set(A),set2(A),removeAll(A,Xb,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_5592_removeAll__id,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ( removeAll(A,Xb,Xs) = Xs ) ) ).

% removeAll_id
tff(fact_5593_length__removeAll__less__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,Xb,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_removeAll_less_eq
tff(fact_5594_length__removeAll__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,Xb,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_removeAll_less
tff(fact_5595_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_ql(nat,fun(nat,$o))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_5596_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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_5597_bit_Oabstract__boolean__algebra__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => boolea2506097494486148201lgebra(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.abstract_boolean_algebra_axioms
tff(fact_5598_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ~ ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),S)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X4)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_5599_subrelI,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( ! [X2: A,Y4: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y4)),R)
         => aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y4)),S3) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),S3) ) ).

% subrelI
tff(fact_5600_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_qm(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_qm(set(product_prod(A,B)),fun(A,fun(B,$o))),S))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S) ) ).

% pred_subset_eq2
tff(fact_5601_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X: A,Xa2: B] :
          ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa2)),R2)
        <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa2)),S) )
    <=> ( R2 = S ) ) ).

% pred_equals_eq2
tff(fact_5602_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),Y: A,F2: fun(A,B)] :
          ( finite_finite2(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(set(A),$o,member(A,Y),S)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))),aa(A,B,F2,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_5603_eq__snd__iff,axiom,
    ! [B: $tType,A: $tType,B2: A,P2: product_prod(B,A)] :
      ( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P2) )
    <=> ? [A5: B] : P2 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),B2) ) ).

% eq_snd_iff
tff(fact_5604_sndI,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),Y: A,Z: B] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z) )
     => ( aa(product_prod(A,B),B,product_snd(A,B),Xb) = Z ) ) ).

% sndI
tff(fact_5605_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A2: A,P2: product_prod(A,B)] :
      ( ( A2 = aa(product_prod(A,B),A,product_fst(A,B),P2) )
    <=> ? [B5: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B5) ) ).

% eq_fst_iff
tff(fact_5606_fstI,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B),Y: A,Z: B] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z) )
     => ( aa(product_prod(A,B),A,product_fst(A,B),Xb) = Y ) ) ).

% fstI
tff(fact_5607_times__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qo(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_5608_Gcd__remove0__nat,axiom,
    ! [M6: set(nat)] :
      ( finite_finite2(nat,M6)
     => ( gcd_Gcd(nat,M6) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M6),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_5609_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_5610_iszero__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: num] :
          ( ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),W)) ) ) ).

% iszero_neg_numeral
tff(fact_5611_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X2: nat,Y4: nat] : Z != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y4)) ).

% eq_Abs_Integ
tff(fact_5612_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_5613_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_5614_Gcd__1,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( aa(set(A),$o,member(A,one_one(A)),A3)
         => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ).

% Gcd_1
tff(fact_5615_Gcd__nat__eq__one,axiom,
    ! [N2: set(nat)] :
      ( aa(set(nat),$o,member(nat,one_one(nat)),N2)
     => ( gcd_Gcd(nat,N2) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_5616_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_5617_not__iszero__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W: num] : ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),W)) ) ).

% not_iszero_numeral
tff(fact_5618_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
        <=> ring_1_iszero(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) ) ) ).

% eq_iff_iszero_diff
tff(fact_5619_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ( aa(set(A),$o,member(A,A2),A3)
           => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_5620_nat_Oabs__eq,axiom,
    ! [Xb: product_prod(nat,nat)] : aa(int,nat,nat2,aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),Xb) ).

% nat.abs_eq
tff(fact_5621_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xb)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_5622_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_5623_not__iszero__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),one2)) ) ).

% not_iszero_Numeral1
tff(fact_5624_not__iszero__neg__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_iszero_neg_1
tff(fact_5625_eq__numeral__iff__iszero_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Xb,Y)) ) ) ).

% eq_numeral_iff_iszero(1)
tff(fact_5626_Gcd__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_Gcd(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),gcd_Gcd(B,aa(set(A),set(B),image(A,B,F2),A3))),gcd_Gcd(B,aa(set(A),set(B),image(A,B,G),A3))) ) ) ).

% Gcd_mono
tff(fact_5627_zero__int__def,axiom,
    zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))) ).

% zero_int_def
tff(fact_5628_int__def,axiom,
    ! [Nb: nat] : aa(nat,int,semiring_1_of_nat(int),Nb) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Nb),zero_zero(nat))) ).

% int_def
tff(fact_5629_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_5630_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xb)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_5631_not__iszero__neg__Numeral1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) ) ).

% not_iszero_neg_Numeral1
tff(fact_5632_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_5633_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_5634_eq__numeral__iff__iszero_I4_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Y,Xb)) ) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_5635_uminus__int_Oabs__eq,axiom,
    ! [Xb: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_qp(nat,fun(nat,product_prod(nat,nat)))),Xb)) ).

% uminus_int.abs_eq
tff(fact_5636_one__int__def,axiom,
    one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).

% one_int_def
tff(fact_5637_of__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: product_prod(nat,nat)] : ring_1_of_int(A,aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_qq(nat,fun(nat,A))),Xb) ) ).

% of_int.abs_eq
tff(fact_5638_less__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_qs(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_5639_less__eq__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_qu(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_5640_eq__numeral__iff__iszero_I6_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,one2,Y)) ) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_5641_eq__numeral__iff__iszero_I5_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: num] :
          ( ( aa(num,A,numeral_numeral(A),Xb) = one_one(A) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Xb,one2)) ) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_5642_plus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_5643_minus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod(nat,nat),Xb: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xaa)),aa(product_prod(nat,nat),int,abs_Integ,Xb)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qy(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_5644_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_5645_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_qz(nat,$o))) ) ).

% semiring_char_def
tff(fact_5646_Gcd__eq__Max,axiom,
    ! [M6: set(nat)] :
      ( finite_finite2(nat,M6)
     => ( ( M6 != bot_bot(set(nat)) )
       => ( ~ aa(set(nat),$o,member(nat,zero_zero(nat)),M6)
         => ( gcd_Gcd(nat,M6) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),aTP_Lamp_ra(nat,set(nat))),M6))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_5647_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      num_of_nat(aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb),inc(num_of_nat(Nb)),one2) ).

% num_of_nat.simps(2)
tff(fact_5648_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_5649_Gcd__int__eq,axiom,
    ! [N2: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),N2)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N2)) ).

% Gcd_int_eq
tff(fact_5650_Gcd__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,aa(set(int),set(int),image(int,int,abs_abs(int)),K5)) = gcd_Gcd(int,K5) ).

% Gcd_abs_eq
tff(fact_5651_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Xb,Y)) = Xb ) ) ) ).

% Inf_atLeastAtMost
tff(fact_5652_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Xb,Y)) = Xb ) ) ) ).

% Inf_atLeastLessThan
tff(fact_5653_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Xb,Y)) = Xb ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_5654_finite__INT,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),I5)
          & finite_finite2(B,aa(A,set(B),A3,X4)) )
     => finite_finite2(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) ) ).

% finite_INT
tff(fact_5655_Gcd__nat__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(nat,aa(set(int),set(nat),image(int,nat,aTP_Lamp_rb(int,nat)),K5)) = aa(int,nat,nat2,gcd_Gcd(int,K5)) ).

% Gcd_nat_abs_eq
tff(fact_5656_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_5657_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S))),A2) ) ) ) ).

% cInf_abs_ge
tff(fact_5658_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),L))),E) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S)),L))),E) ) ) ) ).

% cInf_asclose
tff(fact_5659_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_5660_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),one_one(nat))
     => ( num_of_nat(Nb) = one2 ) ) ).

% num_of_nat_One
tff(fact_5661_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          aa(num,A,numeral_numeral(A),num_of_nat(Nb)) = $ite(Nb = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% numeral_num_of_nat_unfold
tff(fact_5662_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Nb)) = aa(num,num,bit0,num_of_nat(Nb)) ) ) ).

% num_of_nat_double
tff(fact_5663_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(Mb)),num_of_nat(Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_5664_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)) = bot_bot(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X)
             => ? [Xa2: B] :
                  ( aa(set(B),$o,member(B,Xa2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Xa2)),X) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_5665_INF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aTP_Lamp_rc(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% INF_const
tff(fact_5666_ccINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aTP_Lamp_rd(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% ccINF_const
tff(fact_5667_Inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [A3: set(fun(B,A)),Xb: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Inf_Inf(fun(B,A)),A3),Xb) = aa(set(A),A,complete_Inf_Inf(A),aa(set(fun(B,A)),set(A),image(fun(B,A),A,aTP_Lamp_re(B,fun(fun(B,A),A),Xb)),A3)) ) ).

% Inf_apply
tff(fact_5668_INF__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Inf(A)
     => ! [F2: fun(C,fun(B,A)),A3: set(C),Xb: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Inf_Inf(fun(B,A)),aa(set(C),set(fun(B,A)),image(C,fun(B,A),F2),A3)),Xb) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_rf(fun(C,fun(B,A)),fun(B,fun(C,A)),F2),Xb)),A3)) ) ).

% INF_apply
tff(fact_5669_INF__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image(A,A,aTP_Lamp_rg(A,A)),A3)) = aa(set(A),A,complete_Inf_Inf(A),A3) ) ).

% INF_identity_eq
tff(fact_5670_INT__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
    <=> ! [X: B] :
          ( aa(set(B),$o,member(B,X),A3)
         => aa(set(A),$o,member(A,B2),aa(B,set(A),B4,X)) ) ) ).

% INT_iff
tff(fact_5671_INT__I,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B2: B,B4: fun(A,set(B))] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => aa(set(B),$o,member(B,B2),aa(A,set(B),B4,X2)) )
     => aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ).

% INT_I
tff(fact_5672_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A3) = bot_bot(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X)
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_5673_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),S),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).

% Inf_INT_eq2
tff(fact_5674_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_qm(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S)) ) ).

% INF_Int_eq2
tff(fact_5675_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S: set(C),X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),aTP_Lamp_rh(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R)),S)),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image(C,set(product_prod(A,B)),R),S))) ) ).

% INF_INT_eq2
tff(fact_5676_INF__Int__eq,axiom,
    ! [A: $tType,S: set(set(A)),X4: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o))),S)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S)) ) ).

% INF_Int_eq
tff(fact_5677_Inf__set__def,axiom,
    ! [A: $tType,A3: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ri(set(set(A)),fun(A,$o),A3)) ).

% Inf_set_def
tff(fact_5678_INF__INT__eq,axiom,
    ! [A: $tType,B: $tType,R: fun(B,set(A)),S: set(B),X4: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),aTP_Lamp_rj(fun(B,set(A)),fun(B,fun(A,$o)),R)),S)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),R),S))) ) ).

% INF_INT_eq
tff(fact_5679_Inf_OINF__identity__eq,axiom,
    ! [A: $tType,Inf: fun(set(A),A),A3: set(A)] : aa(set(A),A,Inf,aa(set(A),set(A),image(A,A,aTP_Lamp_ml(A,A)),A3)) = aa(set(A),A,Inf,A3) ).

% Inf.INF_identity_eq
tff(fact_5680_Sup_OSUP__identity__eq,axiom,
    ! [A: $tType,Sup: fun(set(A),A),A3: set(A)] : aa(set(A),A,Sup,aa(set(A),set(A),image(A,A,aTP_Lamp_ml(A,A)),A3)) = aa(set(A),A,Sup,A3) ).

% Sup.SUP_identity_eq
tff(fact_5681_Inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [A3: set(fun(A,B)),X4: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Inf_Inf(fun(A,B)),A3),X4) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(A,B)),set(B),image(fun(A,B),B,aTP_Lamp_rk(A,fun(fun(A,B),B),X4)),A3)) ) ).

% Inf_fun_def
tff(fact_5682_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A3)) ) ) ).

% Inf_greatest
tff(fact_5683_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A3))
        <=> ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X) ) ) ) ).

% le_Inf_iff
tff(fact_5684_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V: A] :
          ( aa(set(A),$o,member(A,U),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),V) ) ) ) ).

% Inf_lower2
tff(fact_5685_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A3: set(A)] :
          ( aa(set(A),$o,member(A,Xb),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),Xb) ) ) ).

% Inf_lower
tff(fact_5686_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( ! [B3: A] :
              ( aa(set(A),$o,member(A,B3),B4)
             => ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B3) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4)) ) ) ).

% Inf_mono
tff(fact_5687_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Xb: A] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),I3) )
         => ( ! [Y4: A] :
                ( ! [I: A] :
                    ( aa(set(A),$o,member(A,I),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),I) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A3) = Xb ) ) ) ) ).

% Inf_eqI
tff(fact_5688_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S: set(A),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S)),A2)
        <=> ? [X: A] :
              ( aa(set(A),$o,member(A,X),S)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A2) ) ) ) ).

% Inf_less_iff
tff(fact_5689_INF__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,fun(C,A)),B4: set(C),A3: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aTP_Lamp_rl(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F2),B4)),A3)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,aa(set(B),fun(C,A),aTP_Lamp_rn(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F2),A3)),B4)) ) ).

% INF_commute
tff(fact_5690_INT__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B),A2: B] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
     => ( ~ aa(set(A),$o,member(A,B2),aa(B,set(A),B4,A2))
       => ~ aa(set(B),$o,member(B,A2),A3) ) ) ).

% INT_E
tff(fact_5691_INT__D,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B),A2: B] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
     => ( aa(set(B),$o,member(B,A2),A3)
       => aa(set(A),$o,member(A,B2),aa(B,set(A),B4,A2)) ) ) ).

% INT_D
tff(fact_5692_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),Xb)
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y3)
             => ? [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) ) ) ) ) ).

% Inf_le_iff
tff(fact_5693_INF__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B4: set(B),G: fun(B,C),F2: fun(A,C)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),B4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,X4)),aa(A,C,F2,I3)) ) )
         => ( ! [J2: B] :
                ( aa(set(B),$o,member(B,J2),B4)
               => ? [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A3)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X4)),aa(B,C,G,J2)) ) )
           => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image(A,C,F2),A3)) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image(B,C,G),B4)) ) ) ) ) ).

% INF_eq
tff(fact_5694_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V2: A] :
              ( aa(set(A),$o,member(A,V2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),U) )
         => ( ( A3 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),U) ) ) ) ).

% Inf_less_eq
tff(fact_5695_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4)) ) ) ).

% Inf_superset_mono
tff(fact_5696_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),Xb: B,F2: fun(A,B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),aa(A,B,F2,I3)) )
         => ( ! [Y4: B] :
                ( ! [I: A] :
                    ( aa(set(A),$o,member(A,I),A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y4),aa(A,B,F2,I)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y4),Xb) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3)) = Xb ) ) ) ) ).

% INF_eqI
tff(fact_5697_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B4: set(A),A3: set(B),F2: fun(B,C),G: fun(A,C)] :
          ( ! [M: A] :
              ( aa(set(A),$o,member(A,M),B4)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),A3)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F2,X4)),aa(A,C,G,M)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image(B,C,F2),A3))),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image(A,C,G),B4))) ) ) ).

% INF_mono
tff(fact_5698_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I2: A,A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,I2),A3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(A,B,F2,I2)) ) ) ).

% INF_lower
tff(fact_5699_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A3: set(A)] :
          ( ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,G),A3))) ) ) ).

% INF_mono'
tff(fact_5700_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I2: A,A3: set(A),F2: fun(A,B),U: B] :
          ( aa(set(A),$o,member(A,I2),A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))),U) ) ) ) ).

% INF_lower2
tff(fact_5701_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F2: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)))
        <=> ! [X: B] :
              ( aa(set(B),$o,member(B,X),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F2,X)) ) ) ) ).

% le_INF_iff
tff(fact_5702_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),U: B,F2: fun(A,B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ).

% INF_greatest
tff(fact_5703_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),A2)
        <=> ? [X: B] :
              ( aa(set(B),$o,member(B,X),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),A2) ) ) ) ).

% INF_less_iff
tff(fact_5704_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F2: fun(B,A),A3: set(B),I2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)))
         => ( aa(set(B),$o,member(B,I2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,I2)) ) ) ) ).

% less_INF_D
tff(fact_5705_INT__extend__simps_I10_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,set(A)),F2: fun(B,C),A3: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_ro(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),B4),F2)),A3)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),B4),aa(set(B),set(C),image(B,C,F2),A3))) ).

% INT_extend_simps(10)
tff(fact_5706_INT__lower,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A2),A3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))),aa(A,set(B),B4,A2)) ) ).

% INT_lower
tff(fact_5707_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C5: set(B),B4: fun(A,set(B))] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C5),aa(A,set(B),B4,X2)) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C5),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ).

% INT_greatest
tff(fact_5708_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,X2)),aa(A,set(B),G,X2)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),F2),B4))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),G),A3))) ) ) ).

% INT_anti_mono
tff(fact_5709_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A3: fun(B,set(A)),I5: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5)))
    <=> ! [X: B] :
          ( aa(set(B),$o,member(B,X),I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),aa(B,set(A),A3,X)) ) ) ).

% INT_subset_iff
tff(fact_5710_INT__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B4: fun(B,set(A)),C5: set(B)] : aa(set(A),set(A),insert(A,A2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_rp(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B4)),C5)) ).

% INT_extend_simps(5)
tff(fact_5711_INT__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A3: set(A),A2: B,B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,U),A3)
     => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_rq(B,fun(fun(A,set(B)),fun(A,set(B))),A2),B4)),A3)) = aa(set(B),set(B),insert(B,A2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% INT_insert_distrib
tff(fact_5712_Pow__INT__eq,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : pow2(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) = aa(set(set(set(A))),set(set(A)),complete_Inf_Inf(set(set(A))),aa(set(B),set(set(set(A))),image(B,set(set(A)),aTP_Lamp_rr(fun(B,set(A)),fun(B,set(set(A))),B4)),A3)) ).

% Pow_INT_eq
tff(fact_5713_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),Xb)
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y3)
             => ? [X: B] :
                  ( aa(set(B),$o,member(B,X),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),Y3) ) ) ) ) ).

% INF_le_iff
tff(fact_5714_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),C2: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),C2) )
           => ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),I5)) = C2 )
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),I5)
                 => ( aa(A,B,F2,X) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_5715_INF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B4: set(A),A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),B4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,G),B4))) ) ) ) ).

% INF_superset_mono
tff(fact_5716_cINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),C2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aTP_Lamp_rs(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).

% cINF_const
tff(fact_5717_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,Xb)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_5718_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,Xb)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_5719_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y,Xb)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_5720_INF1__I,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(A,fun(B,$o)),B2: B] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => aa(B,$o,aa(A,fun(B,$o),B4,X2),B2) )
     => aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Inf_Inf(fun(B,$o)),aa(set(A),set(fun(B,$o)),image(A,fun(B,$o),B4),A3)),B2) ) ).

% INF1_I
tff(fact_5721_INF2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A3: set(A),B4: fun(A,fun(B,fun(C,$o))),B2: B,C2: C] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),B4,X2),B2),C2) )
     => aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Inf_Inf(fun(B,fun(C,$o))),aa(set(A),set(fun(B,fun(C,$o))),image(A,fun(B,fun(C,$o)),B4),A3)),B2),C2) ) ).

% INF2_I
tff(fact_5722_INF1__D,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,fun(A,$o)),A3: set(B),B2: A,A2: B] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),B4),A3)),B2)
     => ( aa(set(B),$o,member(B,A2),A3)
       => aa(A,$o,aa(B,fun(A,$o),B4,A2),B2) ) ) ).

% INF1_D
tff(fact_5723_INF1__E,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,fun(A,$o)),A3: set(B),B2: A,A2: B] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),B4),A3)),B2)
     => ( ~ aa(A,$o,aa(B,fun(A,$o),B4,A2),B2)
       => ~ aa(set(B),$o,member(B,A2),A3) ) ) ).

% INF1_E
tff(fact_5724_INF2__D,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,fun(A,fun(B,$o))),A3: set(C),B2: A,C2: B,A2: C] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),B4),A3)),B2),C2)
     => ( aa(set(C),$o,member(C,A2),A3)
       => aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B4,A2),B2),C2) ) ) ).

% INF2_D
tff(fact_5725_INF2__E,axiom,
    ! [B: $tType,A: $tType,C: $tType,B4: fun(C,fun(A,fun(B,$o))),A3: set(C),B2: A,C2: B,A2: C] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),B4),A3)),B2),C2)
     => ( ~ aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B4,A2),B2),C2)
       => ~ aa(set(C),$o,member(C,A2),A3) ) ) ).

% INF2_E
tff(fact_5726_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A2: A] :
        ? [B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B3)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A2) ) ) ).

% ex_gt_or_lt
tff(fact_5727_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,P,A2)
           => ( ~ aa(A,$o,P,B2)
             => ? [C3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
                  & ! [X4: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),C3) )
                     => aa(A,$o,P,X4) )
                  & ! [D6: A] :
                      ( ! [X2: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X2)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),D6) )
                         => aa(A,$o,P,X2) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D6),C3) ) ) ) ) ) ) ).

% complete_interval
tff(fact_5728_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X5: set(A)] :
          ( aa(set(A),$o,member(A,Z),X5)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X2) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X5) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_5729_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X5: set(A),A2: A] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X2) )
         => ( ! [Y4: A] :
                ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X4) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),A2) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X5) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_5730_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ).

% cInf_greatest
tff(fact_5731_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X2) )
           => ( ! [Y4: A] :
                  ( ! [X4: A] :
                      ( aa(set(A),$o,member(A,X4),X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X4) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),A2) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X5) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_5732_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Xb: A] :
          ( finite_finite2(A,X5)
         => ( aa(set(A),$o,member(A,Xb),X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Xb) ) ) ) ).

% cInf_le_finite
tff(fact_5733_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Z)
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z) ) ) ) ) ).

% cInf_lessD
tff(fact_5734_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Xb: A,A2: A] :
          ( finite_finite2(A,X5)
         => ( aa(set(A),$o,member(A,Xb),X5)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X2) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X5)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_5735_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),Mb: B,F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(A,B,F2,X2)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ) ).

% cINF_greatest
tff(fact_5736_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),A2: A] :
          ( finite_finite2(A,X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X5))
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_5737_card__UNION,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( finite_finite2(set(A),A3)
     => ( ! [X2: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X2),A3)
           => finite_finite2(A,X2) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) = aa(int,nat,nat2,aa(set(set(set(A))),int,groups7311177749621191930dd_sum(set(set(A)),int,aTP_Lamp_rt(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_ru(set(set(A)),fun(set(set(A)),$o),A3)))) ) ) ) ).

% card_UNION
tff(fact_5738_INF__filter__not__bot,axiom,
    ! [A: $tType,B: $tType,B4: set(A),F4: fun(A,filter(B))] :
      ( ! [X6: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),B4)
         => ( finite_finite2(A,X6)
           => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),X6)) != bot_bot(filter(B)) ) ) )
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),B4)) != bot_bot(filter(B)) ) ) ).

% INF_filter_not_bot
tff(fact_5739_less__eq__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_qu(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,Xb)),aa(int,product_prod(nat,nat),rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_5740_SUP__identity__eq,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image(A,A,aTP_Lamp_rv(A,A)),A3)) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ).

% SUP_identity_eq
tff(fact_5741_UN__I,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B2: B,B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(set(B),$o,member(B,B2),aa(A,set(B),B4,A2))
       => aa(set(B),$o,member(B,B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% UN_I
tff(fact_5742_UN__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
    <=> ? [X: B] :
          ( aa(set(B),$o,member(B,X),A3)
          & aa(set(A),$o,member(A,B2),aa(B,set(A),B4,X)) ) ) ).

% UN_iff
tff(fact_5743_Sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [A3: set(fun(B,A)),Xb: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),A3),Xb) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,A)),set(A),image(fun(B,A),A,aTP_Lamp_rw(B,fun(fun(B,A),A),Xb)),A3)) ) ).

% Sup_apply
tff(fact_5744_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Xb,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_5745_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,Xb)) = Xb ) ) ) ).

% cSup_atLeastAtMost
tff(fact_5746_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Xb,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_5747_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,Xb)) = Xb ) ) ) ).

% cSup_atLeastLessThan
tff(fact_5748_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Xb,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_5749_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,Xb)) = Xb ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_5750_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A3: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B4),A3)) )
        <=> ! [X: B] :
              ( aa(set(B),$o,member(B,X),A3)
             => ( aa(B,A,B4,X) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_5751_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B4),A3)) = bot_bot(A) )
        <=> ! [X: B] :
              ( aa(set(B),$o,member(B,X),A3)
             => ( aa(B,A,B4,X) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_5752_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_rx(B,A)),A3)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_5753_ccSUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_ry(B,A)),A3)) = bot_bot(A) ) ).

% ccSUP_bot
tff(fact_5754_SUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aTP_Lamp_rc(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% SUP_const
tff(fact_5755_ccSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A3: set(A),F2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aTP_Lamp_rd(B,fun(A,B),F2)),A3)) = F2 ) ) ) ).

% ccSUP_const
tff(fact_5756_cSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),C2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aTP_Lamp_rs(B,fun(A,B),C2)),A3)) = C2 ) ) ) ).

% cSUP_const
tff(fact_5757_UN__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),A3: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_rz(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),C2) ).

% UN_constant
tff(fact_5758_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: fun(A,set(B))] :
      ( finite_finite2(A,A3)
     => ( ! [A4: A] :
            ( aa(set(A),$o,member(A,A4),A3)
           => finite_finite2(B,aa(A,set(B),B4,A4)) )
       => finite_finite2(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% finite_UN_I
tff(fact_5759_SUP__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Sup(A)
     => ! [F2: fun(C,fun(B,A)),A3: set(C),Xb: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),aa(set(C),set(fun(B,A)),image(C,fun(B,A),F2),A3)),Xb) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_sa(fun(C,fun(B,A)),fun(B,fun(C,A)),F2),Xb)),A3)) ) ).

% SUP_apply
tff(fact_5760_ccSUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ).

% ccSUP_empty
tff(fact_5761_UN__singleton,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image(A,set(A),aTP_Lamp_sb(A,set(A))),A3)) = A3 ).

% UN_singleton
tff(fact_5762_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B4: fun(B,set(A)),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_rp(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B4)),C5)) = $ite(C5 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),insert(A,A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5)))) ).

% UN_simps(1)
tff(fact_5763_Compl__UN,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_sc(fun(B,set(A)),fun(B,set(A)),B4)),A3)) ).

% Compl_UN
tff(fact_5764_Compl__INT,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_sc(fun(B,set(A)),fun(B,set(A)),B4)),A3)) ).

% Compl_INT
tff(fact_5765_set__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),set(A),set2(A),concat(A,Xs)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).

% set_concat
tff(fact_5766_UN__extend__simps_I9_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,C5: fun(C,set(A)),B4: fun(B,set(C)),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_sd(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),C5),B4)),A3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),C5),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image(B,set(C),B4),A3)))) ).

% UN_extend_simps(9)
tff(fact_5767_SUP__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,fun(C,A)),B4: set(C),A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aTP_Lamp_se(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F2),B4)),A3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(set(B),fun(C,A),aTP_Lamp_sf(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F2),A3)),B4)) ) ).

% SUP_commute
tff(fact_5768_UN__E,axiom,
    ! [A: $tType,B: $tType,B2: A,B4: fun(B,set(A)),A3: set(B)] :
      ( aa(set(A),$o,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))
     => ~ ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A3)
           => ~ aa(set(A),$o,member(A,B2),aa(B,set(A),B4,X2)) ) ) ).

% UN_E
tff(fact_5769_SUP__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),G: fun(C,set(B)),A3: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image(C,set(B),G),A3)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_sg(fun(B,A),fun(fun(C,set(B)),fun(C,A)),F2),G)),A3)) ) ).

% SUP_UNION
tff(fact_5770_Sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [A3: set(fun(A,B)),X4: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A3),X4) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),image(fun(A,B),B,aTP_Lamp_sh(A,fun(fun(A,B),B),X4)),A3)) ) ).

% Sup_fun_def
tff(fact_5771_UN__UN__flatten,axiom,
    ! [B: $tType,A: $tType,C: $tType,C5: fun(B,set(A)),B4: fun(C,set(B)),A3: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),C5),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image(C,set(B),B4),A3)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_si(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),C5),B4)),A3)) ).

% UN_UN_flatten
tff(fact_5772_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),image(B,set(set(A)),aTP_Lamp_rr(fun(B,set(A)),fun(B,set(set(A))),B4)),A3))),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)))) ).

% UN_Pow_subset
tff(fact_5773_SUP__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B4: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),B4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,I3)),aa(B,C,G,X4)) ) )
         => ( ! [J2: B] :
                ( aa(set(B),$o,member(B,J2),B4)
               => ? [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A3)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,J2)),aa(A,C,F2,X4)) ) )
           => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image(A,C,F2),A3)) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,G),B4)) ) ) ) ) ).

% SUP_eq
tff(fact_5774_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X5))
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X2) ) ) ) ) ).

% less_cSupD
tff(fact_5775_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
         => ( ( X5 != bot_bot(set(A)) )
           => ~ ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),X5)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X2) ) ) ) ) ).

% less_cSupE
tff(fact_5776_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Xb: A,A2: A] :
          ( finite_finite2(A,X5)
         => ( aa(set(A),$o,member(A,Xb),X5)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),A2) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A2) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_5777_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X5: set(A)] :
          ( aa(set(A),$o,member(A,Z),X5)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_5778_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X5: set(A),A2: A] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),A2) )
         => ( ! [Y4: A] :
                ( ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),X5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y4) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_5779_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Xb: A] :
          ( finite_finite2(A,X5)
         => ( aa(set(A),$o,member(A,Xb),X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ).

% le_cSup_finite
tff(fact_5780_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),Z: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X5)),Z) ) ) ) ).

% cSup_least
tff(fact_5781_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X5: set(A),A2: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),X5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),A2) )
           => ( ! [Y4: A] :
                  ( ! [X4: A] :
                      ( aa(set(A),$o,member(A,X4),X5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y4) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y4) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X5) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_5782_card__Union__le__sum__card,axiom,
    ! [A: $tType,U3: set(set(A))] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U3))),aa(set(set(A)),nat,groups7311177749621191930dd_sum(set(A),nat,finite_card(A)),U3)) ).

% card_Union_le_sum_card
tff(fact_5783_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S))
        <=> ? [X: A] :
              ( aa(set(A),$o,member(A,X),S)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X) ) ) ) ).

% less_Sup_iff
tff(fact_5784_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,A3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A3))
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb)
             => ? [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X) ) ) ) ) ).

% le_Sup_iff
tff(fact_5785_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Xb: A] :
          ( ! [Y4: A] :
              ( aa(set(A),$o,member(A,Y4),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
         => ( ! [Y4: A] :
                ( ! [Z2: A] :
                    ( aa(set(A),$o,member(A,Z2),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y4) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y4) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A3) = Xb ) ) ) ) ).

% Sup_eqI
tff(fact_5786_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( ! [A4: A] :
              ( aa(set(A),$o,member(A,A4),A3)
             => ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),B4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X4) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4)) ) ) ).

% Sup_mono
tff(fact_5787_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z) ) ) ).

% Sup_least
tff(fact_5788_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A3: set(A)] :
          ( aa(set(A),$o,member(A,Xb),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ).

% Sup_upper
tff(fact_5789_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2)
        <=> ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B2) ) ) ) ).

% Sup_le_iff
tff(fact_5790_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V: A] :
          ( aa(set(A),$o,member(A,U),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V),U)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ).

% Sup_upper2
tff(fact_5791_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4)) ) ) ).

% Sup_subset_mono
tff(fact_5792_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V2: A] :
              ( aa(set(A),$o,member(A,V2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2) )
         => ( ( A3 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ).

% less_eq_Sup
tff(fact_5793_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I2: A,A3: set(A),U: B,F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,I2),A3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I2))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ) ).

% SUP_upper2
tff(fact_5794_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),U)
        <=> ! [X: B] :
              ( aa(set(B),$o,member(B,X),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X)),U) ) ) ) ).

% SUP_le_iff
tff(fact_5795_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I2: A,A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,I2),A3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ).

% SUP_upper
tff(fact_5796_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A3: set(A)] :
          ( ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G),A3))) ) ) ).

% SUP_mono'
tff(fact_5797_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B),U: B] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),U) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),U) ) ) ).

% SUP_least
tff(fact_5798_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A3: set(A),B4: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [N: A] :
              ( aa(set(A),$o,member(A,N),A3)
             => ? [X4: B] :
                  ( aa(set(B),$o,member(B,X4),B4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,N)),aa(B,C,G,X4)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image(A,C,F2),A3))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,G),B4))) ) ) ).

% SUP_mono
tff(fact_5799_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B),Xb: B] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),Xb) )
         => ( ! [Y4: B] :
                ( ! [I: A] :
                    ( aa(set(A),$o,member(A,I),A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),Y4) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),Y4) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3)) = Xb ) ) ) ) ).

% SUP_eqI
tff(fact_5800_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),Y)
         => ( aa(set(B),$o,member(B,I2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,I2)),Y) ) ) ) ).

% SUP_lessD
tff(fact_5801_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,F2: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)))
        <=> ? [X: B] :
              ( aa(set(B),$o,member(B,X),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F2,X)) ) ) ) ).

% less_SUP_iff
tff(fact_5802_UN__extend__simps_I10_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,set(A)),F2: fun(B,C),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_ro(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),B4),F2)),A3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),B4),aa(set(B),set(C),image(B,C,F2),A3))) ).

% UN_extend_simps(10)
tff(fact_5803_image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),B4: fun(C,set(B)),A3: set(C)] : aa(set(B),set(A),image(B,A,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image(C,set(B),B4),A3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_sj(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),F2),B4)),A3)) ).

% image_UN
tff(fact_5804_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A3: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) = bot_bot(set(A)) )
    <=> ! [X: B] :
          ( aa(set(B),$o,member(B,X),A3)
         => ( aa(B,set(A),B4,X) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_5805_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A3: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) )
    <=> ! [X: B] :
          ( aa(set(B),$o,member(B,X),A3)
         => ( aa(B,set(A),B4,X) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_5806_UN__empty,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_5807_UN__empty2,axiom,
    ! [B: $tType,A: $tType,A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_sk(B,set(A))),A3)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_5808_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I5: set(B),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),B4)
    <=> ! [X: B] :
          ( aa(set(B),$o,member(B,X),I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),A3,X)),B4) ) ) ).

% UN_subset_iff
tff(fact_5809_UN__upper,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A2),A3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B4,A2)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ).

% UN_upper
tff(fact_5810_UN__least,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(A,set(B)),C5: set(B)] :
      ( ! [X2: A] :
          ( aa(set(A),$o,member(A,X2),A3)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B4,X2)),C5) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))),C5) ) ).

% UN_least
tff(fact_5811_UN__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,X2)),aa(A,set(B),G,X2)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),F2),A3))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),G),B4))) ) ) ).

% UN_mono
tff(fact_5812_UN__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A3: set(A),A2: B,B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,U),A3)
     => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_rq(B,fun(fun(A,set(B)),fun(A,set(B))),A2),B4)),A3)) = aa(set(B),set(B),insert(B,A2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% UN_insert_distrib
tff(fact_5813_UN__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C5: set(B),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_sl(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C5)) ).

% UN_extend_simps(6)
tff(fact_5814_INT__extend__simps_I9_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,C5: fun(C,set(A)),B4: fun(B,set(C)),A3: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_sm(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),C5),B4)),A3)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),C5),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image(B,set(C),B4),A3)))) ).

% INT_extend_simps(9)
tff(fact_5815_image__Union,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(set(B))] : aa(set(B),set(A),image(B,A,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F2)),S)) ).

% image_Union
tff(fact_5816_UN__extend__simps_I8_J,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(set(B))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image(set(B),set(A),aTP_Lamp_sn(fun(B,set(A)),fun(set(B),set(A)),B4)),A3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A3))) ).

% UN_extend_simps(8)
tff(fact_5817_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,F2: fun(B,A),A3: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)))
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb)
             => ? [X: B] :
                  ( aa(set(B),$o,member(B,X),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),aa(B,A,F2,X)) ) ) ) ) ).

% le_SUP_iff
tff(fact_5818_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),C2: B,F2: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),aa(A,B,F2,I3)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),I5)) = C2 )
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),I5)
                 => ( aa(A,B,F2,X) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_5819_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),M6: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),M6) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),M6) ) ) ) ).

% cSUP_least
tff(fact_5820_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),A2: A] :
          ( finite_finite2(A,X5)
         => ( ( X5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X5)),A2)
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),X5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A2) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_5821_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ).

% Inf_le_Sup
tff(fact_5822_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S))),A2) ) ) ) ).

% cSup_abs_le
tff(fact_5823_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U3: set(set(A))] :
      ( ! [X2: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X2),U3)
         => finite_finite2(A,X2) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U3))),aa(set(set(A)),nat,groups7311177749621191930dd_sum(set(A),nat,finite_card(A)),U3)) ) ).

% card_Union_le_sum_card_weak
tff(fact_5824_SUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),B4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G),B4))) ) ) ) ).

% SUP_subset_mono
tff(fact_5825_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,A3: set(B)] :
          aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_so(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),bot_bot(A),C2) ) ).

% SUP_constant
tff(fact_5826_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_5827_uminus__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B4: fun(B,A),A3: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,B4),A3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_sp(fun(B,A),fun(B,A),B4)),A3)) ) ).

% uminus_INF
tff(fact_5828_uminus__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [B4: fun(B,A),A3: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B4),A3))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aTP_Lamp_sp(fun(B,A),fun(B,A),B4)),A3)) ) ).

% uminus_SUP
tff(fact_5829_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B4: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),insert(A,A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) = $ite(C5 = bot_bot(set(B)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_rp(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B4)),C5))) ).

% UN_extend_simps(1)
tff(fact_5830_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F2: fun(B,C),A9: fun(A,set(C))] :
      ( ! [I3: A,J2: A] :
          ( aa(set(A),$o,member(A,I3),I5)
         => ( aa(set(A),$o,member(A,J2),I5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,I3)),aa(A,set(B),A3,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I3)) ) ) )
     => ( ! [I3: A] :
            ( aa(set(A),$o,member(A,I3),I5)
           => bij_betw(B,C,F2,aa(A,set(B),A3,I3),aa(A,set(C),A9,I3)) )
       => bij_betw(B,C,F2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image(A,set(C),A9),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_5831_UN__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_sq(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5)) ).

% UN_extend_simps(7)
tff(fact_5832_INT__extend__simps_I8_J,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(set(B))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(B)),set(set(A)),image(set(B),set(A),aTP_Lamp_sr(fun(B,set(A)),fun(set(B),set(A)),B4)),A3)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A3))) ).

% INT_extend_simps(8)
tff(fact_5833_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),L))),E) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S)),L))),E) ) ) ) ).

% cSup_asclose
tff(fact_5834_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ).

% INF_le_SUP
tff(fact_5835_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_ss(fun(B,A),fun(B,set(A)),F2)),A3)) = aa(set(B),set(A),image(B,A,F2),A3) ).

% UNION_singleton_eq_range
tff(fact_5836_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) = $ite(C5 = bot_bot(set(B)),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_sq(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5))) ).

% INT_extend_simps(4)
tff(fact_5837_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_st(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ).

% UN_le_add_shift_strict
tff(fact_5838_UN__le__add__shift,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_st(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K)),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K)))) ).

% UN_le_add_shift
tff(fact_5839_nat_Orep__eq,axiom,
    ! [Xb: int] : aa(int,nat,nat2,Xb) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),aa(int,product_prod(nat,nat),rep_Integ,Xb)) ).

% nat.rep_eq
tff(fact_5840_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( finite_finite2(A,I5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_su(fun(A,set(B)),fun(A,nat),A3)),I5)) ) ).

% card_UN_le
tff(fact_5841_of__int_Orep__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: int] : ring_1_of_int(A,Xb) = aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_qq(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,Xb)) ) ).

% of_int.rep_eq
tff(fact_5842_less__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xb),Xaa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_qs(nat,fun(nat,fun(product_prod(nat,nat),$o)))),aa(int,product_prod(nat,nat),rep_Integ,Xb)),aa(int,product_prod(nat,nat),rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_5843_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,set(A)),G: fun(C,set(B)),Xb: C,X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),aa(C,set(B),G,Xb)))),X5)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(C,set(B),G,Xb)),aa(fun(B,$o),set(B),collect(B),aa(set(A),fun(B,$o),aTP_Lamp_sv(fun(B,set(A)),fun(set(A),fun(B,$o)),F2),X5))) ) ).

% UN_image_subset
tff(fact_5844_length__remdups__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),remdups(A,concat(A,Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))) ).

% length_remdups_concat
tff(fact_5845_UN__UN__split__split__eq,axiom,
    ! [D: $tType,E3: $tType,A: $tType,C: $tType,B: $tType,A3: fun(B,fun(C,fun(D,fun(E3,set(A))))),Y6: set(product_prod(D,E3)),X5: set(product_prod(B,C))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(set(product_prod(D,E3)),fun(B,fun(C,set(A))),aTP_Lamp_sw(fun(B,fun(C,fun(D,fun(E3,set(A))))),fun(set(product_prod(D,E3)),fun(B,fun(C,set(A)))),A3),Y6))),X5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),image(product_prod(B,C),set(A),aa(set(product_prod(D,E3)),fun(product_prod(B,C),set(A)),aTP_Lamp_sz(fun(B,fun(C,fun(D,fun(E3,set(A))))),fun(set(product_prod(D,E3)),fun(product_prod(B,C),set(A))),A3),Y6)),X5)) ).

% UN_UN_split_split_eq
tff(fact_5846_SUP2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A2: A,A3: set(A),B4: fun(A,fun(B,fun(C,$o))),B2: B,C2: C] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),B4,A2),B2),C2)
       => aa(C,$o,aa(B,fun(C,$o),aa(set(fun(B,fun(C,$o))),fun(B,fun(C,$o)),complete_Sup_Sup(fun(B,fun(C,$o))),aa(set(A),set(fun(B,fun(C,$o))),image(A,fun(B,fun(C,$o)),B4),A3)),B2),C2) ) ) ).

% SUP2_I
tff(fact_5847_SUP1__I,axiom,
    ! [A: $tType,B: $tType,A2: A,A3: set(A),B4: fun(A,fun(B,$o)),B2: B] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(B,$o,aa(A,fun(B,$o),B4,A2),B2)
       => aa(B,$o,aa(set(fun(B,$o)),fun(B,$o),complete_Sup_Sup(fun(B,$o)),aa(set(A),set(fun(B,$o)),image(A,fun(B,$o),B4),A3)),B2) ) ) ).

% SUP1_I
tff(fact_5848_set__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),remdups(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_remdups
tff(fact_5849_length__remdups__eq,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),remdups(A,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ( remdups(A,Xs) = Xs ) ) ).

% length_remdups_eq
tff(fact_5850_length__remdups__leq,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_remdups_leq
tff(fact_5851_Inf__real__def,axiom,
    ! [X5: set(real)] : aa(set(real),real,complete_Inf_Inf(real),X5) = aa(real,real,uminus_uminus(real),aa(set(real),real,complete_Sup_Sup(real),aa(set(real),set(real),image(real,real,uminus_uminus(real)),X5))) ).

% Inf_real_def
tff(fact_5852_SUP1__E,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,fun(A,$o)),A3: set(B),B2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),B4),A3)),B2)
     => ~ ! [X2: B] :
            ( aa(set(B),$o,member(B,X2),A3)
           => ~ aa(A,$o,aa(B,fun(A,$o),B4,X2),B2) ) ) ).

% SUP1_E
tff(fact_5853_SUP2__E,axiom,
    ! [A: $tType,C: $tType,B: $tType,B4: fun(C,fun(A,fun(B,$o))),A3: set(C),B2: A,C2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),B4),A3)),B2),C2)
     => ~ ! [X2: C] :
            ( aa(set(C),$o,member(C,X2),A3)
           => ~ aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),B4,X2),B2),C2) ) ) ).

% SUP2_E
tff(fact_5854_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_qm(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S)) ) ).

% SUP_Sup_eq2
tff(fact_5855_SUP__Sup__eq,axiom,
    ! [A: $tType,S: set(set(A)),X4: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o))),S)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S)) ) ).

% SUP_Sup_eq
tff(fact_5856_SUP__UN__eq,axiom,
    ! [A: $tType,B: $tType,R: fun(B,set(A)),S: set(B),X4: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image(B,fun(A,$o),aTP_Lamp_rj(fun(B,set(A)),fun(B,fun(A,$o)),R)),S)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),R),S))) ) ).

% SUP_UN_eq
tff(fact_5857_Sup__set__def,axiom,
    ! [A: $tType,A3: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ta(set(set(A)),fun(A,$o),A3)) ).

% Sup_set_def
tff(fact_5858_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S: set(C),X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),aTP_Lamp_rh(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R)),S)),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image(C,set(product_prod(A,B)),R),S))) ) ).

% SUP_UN_eq2
tff(fact_5859_length__remdups__card__conv,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),remdups(A,Xs)) = aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs)) ).

% length_remdups_card_conv
tff(fact_5860_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),S),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).

% Sup_SUP_eq2
tff(fact_5861_UN__constant__eq,axiom,
    ! [A: $tType,B: $tType,A2: A,A3: set(A),F2: fun(A,set(B)),C2: set(B)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => ( aa(A,set(B),F2,X2) = C2 ) )
       => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),F2),A3)) = C2 ) ) ) ).

% UN_constant_eq
tff(fact_5862_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F2: fun(nat,set(A)),S: set(A)] :
      ( ! [I3: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F2,I3)),S)
     => ( finite_finite2(A,S)
       => ( ? [N6: nat] :
              ( ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N6)
                 => ! [M: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N6)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F2,M)),aa(nat,set(A),F2,N)) ) ) )
              & ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                 => ( aa(nat,set(A),F2,N6) = aa(nat,set(A),F2,N) ) ) )
         => ( aa(nat,set(A),F2,aa(set(A),nat,finite_card(A),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),F2),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_5863_card__partition,axiom,
    ! [A: $tType,C5: set(set(A)),K: nat] :
      ( finite_finite2(set(A),C5)
     => ( finite_finite2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5))
       => ( ! [C3: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),C3),C5)
             => ( aa(set(A),nat,finite_card(A),C3) = K ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),C1),C5)
               => ( aa(set(set(A)),$o,member(set(A),C22),C5)
                 => ( ( C1 != C22 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(set(A)),nat,finite_card(set(A)),C5)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5)) ) ) ) ) ) ).

% card_partition
tff(fact_5864_suminf__eq__SUP__real,axiom,
    ! [X5: fun(nat,real)] :
      ( summable(real,X5)
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,X5,I3))
       => ( suminf(real,X5) = aa(set(real),real,complete_Sup_Sup(real),aa(set(nat),set(real),image(nat,real,aTP_Lamp_tb(fun(nat,real),fun(nat,real),X5)),top_top(set(nat)))) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_5865_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% inf.bounded_iff
tff(fact_5866_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) ) ) ) ).

% le_inf_iff
tff(fact_5867_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite2(option(A),top_top(set(option(A))))
    <=> finite_finite2(A,top_top(set(A))) ) ).

% finite_option_UNIV
tff(fact_5868_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      aa(fun(A,$o),set(A),collect(A),aTP_Lamp_tc($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_5869_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),P))
     => ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fy(fun(A,$o),fun(A,$o),P)))
      <=> finite_finite2(A,top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_5870_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_5871_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_5872_range__diff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% range_diff
tff(fact_5873_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A3) = top_top(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),top_top(A))
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa2) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_5874_Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_5875_Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_5876_surj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( ( aa(set(A),set(A),image(A,A,F2),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_5877_Gcd__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).

% Gcd_UNIV
tff(fact_5878_Diff__Compl,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ).

% Diff_Compl
tff(fact_5879_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_5880_INF__top__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A3: set(B)] :
          ( ( top_top(A) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,B4),A3)) )
        <=> ! [X: B] :
              ( aa(set(B),$o,member(B,X),A3)
             => ( aa(B,A,B4,X) = top_top(A) ) ) ) ) ).

% INF_top_conv(2)
tff(fact_5881_INF__top__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,B4),A3)) = top_top(A) )
        <=> ! [X: B] :
              ( aa(set(B),$o,member(B,X),A3)
             => ( aa(B,A,B4,X) = top_top(A) ) ) ) ) ).

% INF_top_conv(1)
tff(fact_5882_INF__top,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aTP_Lamp_td(B,A)),A3)) = top_top(A) ) ).

% INF_top
tff(fact_5883_ccINF__top,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aTP_Lamp_te(B,A)),A3)) = top_top(A) ) ).

% ccINF_top
tff(fact_5884_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)) = top_top(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),top_top(A))
             => ? [Xa2: B] :
                  ( aa(set(B),$o,member(B,Xa2),A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(B,A,F2,Xa2)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_5885_range__constant,axiom,
    ! [B: $tType,A: $tType,Xb: A] : aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_mw(A,fun(B,A)),Xb)),top_top(set(B))) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) ).

% range_constant
tff(fact_5886_ccINF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ).

% ccINF_empty
tff(fact_5887_INT__constant,axiom,
    ! [B: $tType,A: $tType,C2: set(A),A3: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_rz(set(A),fun(B,set(A)),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(set(A)),C2) ).

% INT_constant
tff(fact_5888_INT__insert,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A2: B,A3: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),aa(set(B),set(B),insert(B,A2),A3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B4,A2)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) ).

% INT_insert
tff(fact_5889_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_lessThan(A),Xb)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_5890_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),P: fun(A,$o),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tf(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F2)),A3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_5891_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),F2: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_tg(fun(A,B),fun(fun(A,$o),fun(A,B)),F2),P)),A3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_5892_INT__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_th(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5)) = $ite(C5 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5)))) ).

% INT_simps(2)
tff(fact_5893_INT__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B4: set(A),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ti(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C5)) = $ite(C5 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),B4)) ).

% INT_simps(1)
tff(fact_5894_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B4: set(A),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_sl(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C5)) = $ite(C5 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),B4)) ).

% INT_simps(3)
tff(fact_5895_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_sq(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5)) = $ite(C5 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5)))) ).

% INT_simps(4)
tff(fact_5896_sum__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A3: set(A),P: fun(A,$o)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_tj(fun(A,$o),fun(A,B),P)),A3) = aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_5897_sums__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] : aa(A,$o,sums(A,F2),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_tk(fun(nat,A),fun(nat,A),F2)),top_top(set(nat))))) ) ).

% sums_SUP
tff(fact_5898_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S3: set(A),Ta: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(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)),Ta)) ) ).

% translation_Int
tff(fact_5899_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),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_5900_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_tl(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_5901_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F2: fun(B,A)] :
      ( aa(set(A),$o,member(A,B2),aa(set(B),set(A),image(B,A,F2),top_top(set(B))))
     => ~ ! [X2: B] : B2 != aa(B,A,F2,X2) ) ).

% rangeE
tff(fact_5902_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ).

% Sup_inter_less_eq
tff(fact_5903_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite2(fun(A,B),top_top(set(fun(A,B))))
     => ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => finite_finite2(A,top_top(set(A))) ) ) ).

% finite_fun_UNIVD1
tff(fact_5904_Int__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_tm(set(A),fun(set(A),fun(A,$o)),A3),B4)) ).

% Int_def
tff(fact_5905_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_tn(A,$o)) ).

% UNIV_def
tff(fact_5906_Int__Collect,axiom,
    ! [A: $tType,Xb: A,A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,member(A,Xb),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))
    <=> ( aa(set(A),$o,member(A,Xb),A3)
        & aa(A,$o,P,Xb) ) ) ).

% Int_Collect
tff(fact_5907_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_conj_eq
tff(fact_5908_diff__eq,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ).

% diff_eq
tff(fact_5909_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.strict_coboundedI2
tff(fact_5910_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.strict_coboundedI1
tff(fact_5911_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_5912_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C2) ) ) ) ).

% inf.strict_boundedE
tff(fact_5913_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_5914_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb3
tff(fact_5915_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xb: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% less_infI2
tff(fact_5916_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% less_infI1
tff(fact_5917_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( ( A2 != top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_5918_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A2) ) ).

% top.extremum_strict
tff(fact_5919_Diff__Int__distrib2,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C5)) ).

% Diff_Int_distrib2
tff(fact_5920_Diff__Int__distrib,axiom,
    ! [A: $tType,C5: set(A),A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),B4)) ).

% Diff_Int_distrib
tff(fact_5921_Diff__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) ).

% Diff_Diff_Int
tff(fact_5922_Diff__Int2,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5)),B4) ).

% Diff_Int2
tff(fact_5923_Int__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),C5)) ).

% Int_Diff
tff(fact_5924_inf__top_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => semila1105856199041335345_order(A,inf_inf(A),top_top(A),ord_less_eq(A),ord_less(A)) ) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_5925_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_5926_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_5927_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),top_top(A)) ) ).

% top_greatest
tff(fact_5928_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.coboundedI2
tff(fact_5929_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2) ) ) ).

% inf.coboundedI1
tff(fact_5930_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_5931_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb_iff1
tff(fact_5932_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),B2) ) ).

% inf.cobounded2
tff(fact_5933_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),A2) ) ).

% inf.cobounded1
tff(fact_5934_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.order_iff
tff(fact_5935_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) ) ) ) ).

% inf_greatest
tff(fact_5936_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ) ) ).

% inf.boundedI
tff(fact_5937_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% inf.boundedE
tff(fact_5938_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_5939_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = Xb ) ) ) ).

% inf_absorb1
tff(fact_5940_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_5941_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb1
tff(fact_5942_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = Xb ) ) ) ).

% le_iff_inf
tff(fact_5943_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X2: A,Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X2),Y4)),X2)
         => ( ! [X2: A,Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X2),Y4)),Y4)
           => ( ! [X2: A,Y4: A,Z4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F2,Y4),Z4)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F2,Xb),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_5944_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% inf.orderI
tff(fact_5945_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.orderE
tff(fact_5946_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xb: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% le_infI2
tff(fact_5947_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xb) ) ) ).

% le_infI1
tff(fact_5948_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D2)) ) ) ) ).

% inf_mono
tff(fact_5949_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) ) ) ) ).

% le_infI
tff(fact_5950_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2) ) ) ) ).

% le_infE
tff(fact_5951_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Y) ) ).

% inf_le2
tff(fact_5952_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Xb) ) ).

% inf_le1
tff(fact_5953_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Xb) ) ).

% inf_sup_ord(1)
tff(fact_5954_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_5955_Diff__triv,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = A3 ) ) ).

% Diff_triv
tff(fact_5956_Int__Diff__disjoint,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_5957_Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B4)) ).

% Diff_eq
tff(fact_5958_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B4: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,G),B4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_tp(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B4)),A3)) ) ).

% SUP_inf_distrib2
tff(fact_5959_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,F2: fun(B,A),B4: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),B4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tq(A,fun(fun(B,A),fun(B,A)),A2),F2)),B4)) ) ).

% inf_SUP
tff(fact_5960_Sup__inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B4)),A2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),image(A,A,aTP_Lamp_tr(A,fun(A,A),A2)),B4)) ) ).

% Sup_inf
tff(fact_5961_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),B4: set(B),A2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),B4))),A2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_ts(fun(B,A),fun(A,fun(B,A)),F2),A2)),B4)) ) ).

% SUP_inf
tff(fact_5962_INF__absorb,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [K: A,I5: set(A),A3: fun(A,B)] :
          ( aa(set(A),$o,member(A,K),I5)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,A3,K)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,A3),I5))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,A3),I5)) ) ) ) ).

% INF_absorb
tff(fact_5963_INF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,G),A3))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tt(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3)) ) ).

% INF_inf_distrib
tff(fact_5964_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A3) ).

% Compl_eq_Diff_UNIV
tff(fact_5965_bij__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( bij_betw(A,A,F2,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_5966_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S3: set(A),Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),A2)),S3)),aa(set(A),set(A),image(A,A,aTP_Lamp_nr(A,fun(A,A),A2)),Ta)) ) ).

% translation_subtract_Int
tff(fact_5967_SUP__INF,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_tv(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(fun(B,C)),set(A),image(fun(B,C),A,aTP_Lamp_tx(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% SUP_INF
tff(fact_5968_INF__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aTP_Lamp_ty(fun(C,fun(B,A)),fun(B,A),P)),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,C)),set(A),image(fun(B,C),A,aTP_Lamp_tz(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% INF_SUP
tff(fact_5969_UN__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_th(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5)) ).

% UN_extend_simps(5)
tff(fact_5970_UN__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C5: set(B),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ti(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C5)) ).

% UN_extend_simps(4)
tff(fact_5971_Int__UN__distrib,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A3: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_th(set(A),fun(fun(B,set(A)),fun(B,set(A))),B4),A3)),I5)) ).

% Int_UN_distrib
tff(fact_5972_Int__UN__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B),B4: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),B4),J4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_ub(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A3),B4),J4)),I5)) ).

% Int_UN_distrib2
tff(fact_5973_INT__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,K),I5)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,K)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)) ) ) ).

% INT_absorb
tff(fact_5974_INT__Int__distrib,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B4: fun(B,set(A)),I5: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_uc(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),I5))) ).

% INT_Int_distrib
tff(fact_5975_Int__Inter__image,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B4: fun(B,set(A)),C5: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_uc(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) ).

% Int_Inter_image
tff(fact_5976_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F2: fun(A,C)] :
      ( finite_finite2(A,aa(set(B),set(A),image(B,A,G),top_top(set(B))))
     => finite_finite2(C,aa(set(B),set(C),image(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_ud(fun(B,A),fun(fun(A,C),fun(B,C)),G),F2)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_5977_Int__Union,axiom,
    ! [A: $tType,A3: set(A),B4: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3)),B4)) ).

% Int_Union
tff(fact_5978_Int__Union2,axiom,
    ! [A: $tType,B4: set(set(A)),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)),A3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aTP_Lamp_ue(set(A),fun(set(A),set(A)),A3)),B4)) ).

% Int_Union2
tff(fact_5979_INTER__UNIV__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A3: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) = top_top(set(A)) )
    <=> ! [X: B] :
          ( aa(set(B),$o,member(B,X),A3)
         => ( aa(B,set(A),B4,X) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(2)
tff(fact_5980_INTER__UNIV__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A)),A3: set(B)] :
      ( ( top_top(set(A)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) )
    <=> ! [X: B] :
          ( aa(set(B),$o,member(B,X),A3)
         => ( aa(B,set(A),B4,X) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(1)
tff(fact_5981_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% inf_shunt
tff(fact_5982_UN__lessThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_lessThan_UNIV
tff(fact_5983_UN__atMost__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atMost_UNIV
tff(fact_5984_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C2: A,A3: set(B)] :
          aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aTP_Lamp_so(A,fun(B,A),C2)),A3)) = $ite(A3 = bot_bot(set(B)),top_top(A),C2) ) ).

% INF_constant
tff(fact_5985_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_5986_INF__inf__const2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),Xb: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_uf(fun(A,B),fun(B,fun(A,B)),F2),Xb)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),I5))),Xb) ) ) ) ).

% INF_inf_const2
tff(fact_5987_INF__inf__const1,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),Xb: B,F2: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ug(B,fun(fun(A,B),fun(A,B)),Xb),F2)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Xb),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_5988_INF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A2: B,A3: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),insert(B,A2),A3))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,A2)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))) ) ).

% INF_insert
tff(fact_5989_sum_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),B4: set(A)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(set(A),fun(A,B),aTP_Lamp_uh(fun(A,B),fun(set(A),fun(A,B)),G),B4)),A3) ) ) ) ).

% sum.inter_restrict
tff(fact_5990_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),B4: set(A)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_ui(fun(A,B),fun(set(A),fun(A,B)),G),B4)),A3) ) ) ) ).

% prod.inter_restrict
tff(fact_5991_INT__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C5: set(B),B4: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),B4) = $ite(C5 = bot_bot(set(B)),B4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ti(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C5))) ).

% INT_extend_simps(1)
tff(fact_5992_INT__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) = $ite(C5 = bot_bot(set(B)),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_th(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5))) ).

% INT_extend_simps(2)
tff(fact_5993_finite__range__Some,axiom,
    ! [A: $tType] :
      ( finite_finite2(option(A),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A))))
    <=> finite_finite2(A,top_top(set(A))) ) ).

% finite_range_Some
tff(fact_5994_notin__range__Some,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ~ aa(set(option(A)),$o,member(option(A),Xb),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A))))
    <=> ( Xb = none(A) ) ) ).

% notin_range_Some
tff(fact_5995_INT__empty,axiom,
    ! [B: $tType,A: $tType,B4: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_5996_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B11: set(set(A)),A3: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)),A3) = $ite(B11 = bot_bot(set(set(A))),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aTP_Lamp_ue(set(A),fun(set(A),set(A)),A3)),B11))) ).

% Int_Inter_eq(2)
tff(fact_5997_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,A3: set(A),B11: set(set(A))] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)) = $ite(B11 = bot_bot(set(set(A))),A3,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3)),B11))) ).

% Int_Inter_eq(1)
tff(fact_5998_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( finite_finite2(A,S)
           => ( ! [I3: A] :
                  ( aa(set(A),$o,member(A,I3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,H,I3) = zero_zero(B) ) )
             => ( ! [I3: A] :
                    ( aa(set(A),$o,member(A,I3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2))
                   => ( aa(A,B,G,I3) = zero_zero(B) ) )
               => ( ! [X2: A] :
                      ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2))
                     => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
                 => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),T2) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5999_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A,Xb: A] :
          aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),K),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_6000_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),B4: set(A)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))) ) ) ) ).

% sum.Int_Diff
tff(fact_6001_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),B4: set(A)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))) ) ) ) ).

% prod.Int_Diff
tff(fact_6002_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T2)
         => ( finite_finite2(A,S)
           => ( ! [I3: A] :
                  ( aa(set(A),$o,member(A,I3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
                 => ( aa(A,B,H,I3) = one_one(B) ) )
             => ( ! [I3: A] :
                    ( aa(set(A),$o,member(A,I3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2))
                   => ( aa(A,B,G,I3) = one_one(B) ) )
               => ( ! [X2: A] :
                      ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2))
                     => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T2) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_6003_card__Diff__subset__Int,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ).

% card_Diff_subset_Int
tff(fact_6004_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( finite_finite2(A,top_top(set(A)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_6005_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_uj(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% sum.If_cases
tff(fact_6006_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_uk(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% prod.If_cases
tff(fact_6007_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),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_6008_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C5: set(B),B4: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),B4) = $ite(C5 = bot_bot(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),B4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_sl(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C5))) ).

% INT_extend_simps(3)
tff(fact_6009_bij__image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),B4: fun(C,set(A)),A3: set(C)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image(A,B,F2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),B4),A3))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_ul(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F2),B4)),A3)) ) ) ).

% bij_image_INT
tff(fact_6010_UN__UN__finite__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aTP_Lamp_um(fun(nat,set(A)),fun(nat,set(A)),A3)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_6011_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A3: set(A),F2: fun(A,B),B2: B] :
          ( finite_finite2(A,A3)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(B,fun(A,B),aTP_Lamp_un(fun(A,B),fun(B,fun(A,B)),F2),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_uo(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_up(fun(A,B),fun(B,fun(A,$o)),F2),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_6012_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( finite_finite2(A,aa(set(B),set(A),image(B,A,F2),top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image(B,A,F2),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_6013_sum_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [I5: set(A),A3: fun(A,set(B)),G: fun(B,C)] :
          ( finite_finite2(A,I5)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),I5)
               => finite_finite2(B,aa(A,set(B),A3,X2)) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),I5)
                 => ! [Xa3: A] :
                      ( aa(set(A),$o,member(A,Xa3),I5)
                     => ( ( X2 != Xa3 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X2)),aa(A,set(B),A3,Xa3)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_uq(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G)),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_6014_prod_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [I5: set(A),A3: fun(A,set(B)),G: fun(B,C)] :
          ( finite_finite2(A,I5)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),I5)
               => finite_finite2(B,aa(A,set(B),A3,X2)) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),I5)
                 => ! [Xa3: A] :
                      ( aa(set(A),$o,member(A,Xa3),I5)
                     => ( ( X2 != Xa3 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X2)),aa(A,set(B),A3,Xa3)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_ur(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A3),G)),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_6015_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys3: list(A)] :
            ( aa(set(list(A)),$o,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys3) )
       => ( ! [Ys3: list(A),Zs: list(A)] :
              ( aa(set(list(A)),$o,member(list(A),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( aa(set(list(A)),$o,member(list(A),Zs),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys3 != Zs )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_6016_UN__finite__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),C5: set(A)] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),C5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat))))),C5) ) ).

% UN_finite_subset
tff(fact_6017_UN__finite2__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_6018_suminf__eq__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] : suminf(A,F2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_tk(fun(nat,A),fun(nat,A),F2)),top_top(set(nat)))) ) ).

% suminf_eq_SUP
tff(fact_6019_range__mod,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_us(nat,fun(nat,nat),Nb)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ) ).

% range_mod
tff(fact_6020_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( finite_finite2(A,I5)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),I5)
           => finite_finite2(B,aa(A,set(B),A3,X2)) )
       => ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),I5)
             => ! [Xa3: A] :
                  ( aa(set(A),$o,member(A,Xa3),I5)
                 => ( ( X2 != Xa3 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A3,X2)),aa(A,set(B),A3,Xa3)) = bot_bot(set(B)) ) ) ) )
         => ( aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_su(fun(A,set(B)),fun(A,nat),A3)),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_6021_UN__finite2__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B4),top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_6022_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B4: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_ut(A,fun(nat,A),B4)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B4) ) ).

% INF_nat_binary
tff(fact_6023_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys)) = aa(nat,nat,binomial(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))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_6024_Gcd__int__def,axiom,
    ! [K5: set(int)] : gcd_Gcd(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,aa(set(int),set(nat),image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),K5))) ).

% Gcd_int_def
tff(fact_6025_card__UNIV__unit,axiom,
    aa(set(product_unit),nat,finite_card(product_unit),top_top(set(product_unit))) = one_one(nat) ).

% card_UNIV_unit
tff(fact_6026_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_uu($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_6027_card__UNIV__bool,axiom,
    aa(set($o),nat,finite_card($o),top_top(set($o))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% card_UNIV_bool
tff(fact_6028_range__mult,axiom,
    ! [A2: real] :
      aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = $ite(A2 = zero_zero(real),aa(set(real),set(real),insert(real,zero_zero(real)),bot_bot(set(real))),top_top(set(real))) ).

% range_mult
tff(fact_6029_inf__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B4))) ).

% inf_set_def
tff(fact_6030_inf__Int__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A),X4: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),R2)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R2),S)) ) ).

% inf_Int_eq
tff(fact_6031_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B)),X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_qm(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_qm(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R2),S)) ) ).

% inf_Int_eq2
tff(fact_6032_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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_6033_funpow__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ).

% funpow_add
tff(fact_6034_comp__funpow,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F2: fun(B,B)] : aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aa(nat,fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),compow(fun(fun(A,B),fun(A,B))),Nb),comp(B,B,A,F2)) = comp(B,B,A,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Nb),F2)) ).

% comp_funpow
tff(fact_6035_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,F2),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ).

% funpow.simps(2)
tff(fact_6036_funpow__Suc__right,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Nb)),F2) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)),F2) ).

% funpow_Suc_right
tff(fact_6037_length__shuffles,axiom,
    ! [A: $tType,Zs2: list(A),Xs: list(A),Ys: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),Zs2),shuffles(A,Xs,Ys))
     => ( aa(list(A),nat,size_size(list(A)),Zs2) = 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_shuffles
tff(fact_6038_INF__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: fun($o,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set($o),set(A),image($o,A,A3),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa($o,A,A3,$true)),aa($o,A,A3,$false)) ) ).

% INF_UNIV_bool_expand
tff(fact_6039_INT__bool__eq,axiom,
    ! [A: $tType,A3: fun($o,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set($o),set(set(A)),image($o,set(A),A3),top_top(set($o)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa($o,set(A),A3,$true)),aa($o,set(A),A3,$false)) ).

% INT_bool_eq
tff(fact_6040_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) )
         => ( ! [X2: B,Y4: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X2)),aa(B,A,H,Y4))
           => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(fun(C,B),fun(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_6041_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),H: fun(A,B),G: fun(B,C)] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A,Y4: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => ( aa(set(A),$o,member(A,Y4),A3)
                 => ( ( X2 != Y4 )
                   => ( ( aa(A,B,H,X2) = aa(A,B,H,Y4) )
                     => ( aa(B,C,G,aa(A,B,H,X2)) = one_one(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(A),set(B),image(A,B,H),A3)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),H)),A3) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_6042_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F4: fun(A,filter(B))] :
      ( ! [I3: A] :
          ( aa(set(A),$o,member(A,I3),I5)
         => ! [J2: A] :
              ( aa(set(A),$o,member(A,J2),I5)
             => ? [X4: A] :
                  ( aa(set(A),$o,member(A,X4),I5)
                  & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I3)),aa(A,filter(B),F4,J2))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),I5)) = bot_bot(filter(B)) )
      <=> ? [X: A] :
            ( aa(set(A),$o,member(A,X),I5)
            & ( aa(A,filter(B),F4,X) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_6043_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A)] :
      ( distinct(A,Xs)
     => ( distinct(A,Ys)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
         => ( aa(set(list(A)),$o,member(list(A),Zs2),shuffles(A,Xs,Ys))
           => distinct(A,Zs2) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_6044_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),F2: fun(A,C)] :
          ( finite_finite2(A,I5)
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G,aa(A,C,F2,I3))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),aa(set(A),set(C),image(A,C,F2),I5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,G),F2)),I5)) ) ) ) ).

% sum_image_le
tff(fact_6045_cclfp__def,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(A,A)] : order_532582986084564980_cclfp(A,F2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_uv(fun(A,A),fun(nat,A),F2)),top_top(set(nat)))) ) ).

% cclfp_def
tff(fact_6046_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa(real,real,root(Nb),Xb) = $ite(Nb = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_uw(nat,fun(real,real),Nb),Xb)) ).

% root_def
tff(fact_6047_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F2: fun(A,C),G: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,aTP_Lamp_ux(C,set(B))),F2) = aa(fun(A,set(D)),fun(A,set(B)),comp(set(D),set(B),A,image(D,B,G)),aTP_Lamp_uy(A,set(D))) ).

% empty_natural
tff(fact_6048_Union__natural,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : aa(fun(set(set(A)),set(set(B))),fun(set(set(A)),set(B)),comp(set(set(B)),set(B),set(set(A)),complete_Sup_Sup(set(B))),image(set(A),set(B),image(A,B,F2))) = aa(fun(set(set(A)),set(A)),fun(set(set(A)),set(B)),comp(set(A),set(B),set(set(A)),image(A,B,F2)),complete_Sup_Sup(set(A))) ).

% Union_natural
tff(fact_6049_in__Union__o__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xb: A,Gset: fun(B,set(set(A))),Gmap: fun(C,B),A3: C] :
      ( aa(set(A),$o,member(A,Xb),aa(C,set(A),aa(fun(C,B),fun(C,set(A)),comp(B,set(A),C,aa(fun(B,set(set(A))),fun(B,set(A)),comp(set(set(A)),set(A),B,complete_Sup_Sup(set(A))),Gset)),Gmap),A3))
     => aa(set(A),$o,member(A,Xb),aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),comp(set(set(A)),set(A),C,complete_Sup_Sup(set(A))),aa(fun(C,B),fun(C,set(set(A))),comp(B,set(set(A)),C,Gset),Gmap)),A3)) ) ).

% in_Union_o_assoc
tff(fact_6050_card_Ocomp__fun__commute__on,axiom,
    aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) = aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) ).

% card.comp_fun_commute_on
tff(fact_6051_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_uz(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_6052_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_va(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_6053_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_uz(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_6054_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_va(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_6055_the__inv__into__def,axiom,
    ! [B: $tType,A: $tType,A3: set(B),F2: fun(B,A),X4: A] : the_inv_into(B,A,A3,F2,X4) = the(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_vb(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A3),F2),X4)) ).

% the_inv_into_def
tff(fact_6056_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_6057_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_6058_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_6059_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_6060_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_6061_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_6062_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_6063_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,K: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_6064_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se5641148757651400278ts_bit(A,bit_se4197421643247451524op_bit(A,Nb,A2)) = aa(fun(nat,nat),fun(nat,$o),comp(nat,$o,nat,bit_se5641148757651400278ts_bit(A,A2)),aa(nat,fun(nat,nat),plus_plus(nat),Nb)) ) ).

% bit_drop_bit_eq
tff(fact_6065_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),comp(A,A,nat,inverse_inverse(A)),F2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_vc(fun(nat,A),fun(A,fun(nat,A)),F2),C2)) ) ) ).

% summable_inverse_divide
tff(fact_6066_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F2: fun(D,fun(C,A)),G: fun(B,D),Xb: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(fun(B,D),fun(B,fun(C,A)),comp(D,fun(C,A),B,F2),G)),Xb) = aa(C,A,aa(D,fun(C,A),F2,aa(B,D,G,aa(product_prod(B,C),B,product_fst(B,C),Xb))),aa(product_prod(B,C),C,product_snd(B,C),Xb)) ).

% case_prod_comp
tff(fact_6067_prod_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B4: set(set(A)),G: fun(A,B)] :
          ( ! [X2: set(A)] :
              ( aa(set(set(A)),$o,member(set(A),X2),B4)
             => finite_finite2(A,X2) )
         => ( ! [A13: set(A)] :
                ( aa(set(set(A)),$o,member(set(A),A13),B4)
               => ! [A24: set(A)] :
                    ( aa(set(set(A)),$o,member(set(A),A24),B4)
                   => ( ( A13 != A24 )
                     => ! [X2: A] :
                          ( aa(set(A),$o,member(A,X2),A13)
                         => ( aa(set(A),$o,member(A,X2),A24)
                           => ( aa(A,B,G,X2) = one_one(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B4)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),B4) ) ) ) ) ).

% prod.Union_comp
tff(fact_6068_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A),comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_vd(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_6069_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A),comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_ve(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_6070_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_6071_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_6072_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_6073_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_6074_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_6075_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_6076_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_vf(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_6077_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_vf(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_6078_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_vf(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_6079_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_vf(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_6080_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_6081_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),Mb))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_6082_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        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),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          L = zero_zero(code_integer),
          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),zero_zero(code_integer)),K),
          aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),
            $ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(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_vg(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_6083_card__UNIV__char,axiom,
    aa(set(char),nat,finite_card(char),top_top(set(char))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))) ).

% card_UNIV_char
tff(fact_6084_these__insert__Some,axiom,
    ! [A: $tType,Xb: A,A3: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),aa(A,option(A),some(A),Xb)),A3)) = aa(set(A),set(A),insert(A,Xb),these(A,A3)) ).

% these_insert_Some
tff(fact_6085_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_6086_these__image__Some__eq,axiom,
    ! [A: $tType,A3: set(A)] : these(A,aa(set(A),set(option(A)),image(A,option(A),some(A)),A3)) = A3 ).

% these_image_Some_eq
tff(fact_6087_these__insert__None,axiom,
    ! [A: $tType,A3: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),none(A)),A3)) = these(A,A3) ).

% these_insert_None
tff(fact_6088_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),Q: fun(B,$o),G: fun(A,B),X4: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),comp(B,$o,A,aa(fun(B,$o),fun(B,$o),aTP_Lamp_vh(fun(B,$o),fun(fun(B,$o),fun(B,$o)),P),Q)),G),X4)
    <=> ( aa(A,$o,aa(fun(A,B),fun(A,$o),comp(B,$o,A,P),G),X4)
        & aa(A,$o,aa(fun(A,B),fun(A,$o),comp(B,$o,A,Q),G),X4) ) ) ).

% conj_comp_iff
tff(fact_6089_in__these__eq,axiom,
    ! [A: $tType,Xb: A,A3: set(option(A))] :
      ( aa(set(A),$o,member(A,Xb),these(A,A3))
    <=> aa(set(option(A)),$o,member(option(A),aa(A,option(A),some(A),Xb)),A3) ) ).

% in_these_eq
tff(fact_6090_Option_Othese__def,axiom,
    ! [A: $tType,A3: set(option(A))] : these(A,A3) = aa(set(option(A)),set(A),image(option(A),A,the2(A)),aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_vi(set(option(A)),fun(option(A),$o),A3))) ).

% Option.these_def
tff(fact_6091_these__not__empty__eq,axiom,
    ! [A: $tType,B4: set(option(A))] :
      ( ( these(A,B4) != bot_bot(set(A)) )
    <=> ( ( B4 != bot_bot(set(option(A))) )
        & ( B4 != aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_6092_these__empty__eq,axiom,
    ! [A: $tType,B4: set(option(A))] :
      ( ( these(A,B4) = bot_bot(set(A)) )
    <=> ( ( B4 = bot_bot(set(option(A))) )
        | ( B4 = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_6093_Some__image__these__eq,axiom,
    ! [A: $tType,A3: set(option(A))] : aa(set(A),set(option(A)),image(A,option(A),some(A)),these(A,A3)) = aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_vi(set(option(A)),fun(option(A),$o),A3)) ).

% Some_image_these_eq
tff(fact_6094_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),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_6095_char__of__quasi__inj,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: A,Nb: A] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),Mb) = aa(A,char,unique5772411509450598832har_of(A),Nb) )
        <=> ( modulo_modulo(A,Mb,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) = modulo_modulo(A,Nb,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ) ) ) ).

% char_of_quasi_inj
tff(fact_6096_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A] : aa(A,char,unique5772411509450598832har_of(A),modulo_modulo(A,Nb,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) = aa(A,char,unique5772411509450598832har_of(A),Nb) ) ).

% char_of_mod_256
tff(fact_6097_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,Mb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))),Nb)
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),Mb)) = aa(A,char,unique5772411509450598832har_of(A),Mb) ) ) ) ).

% char_of_take_bit_eq
tff(fact_6098_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),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ) ).

% of_char_of
tff(fact_6099_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A] : aa(A,char,unique5772411509450598832har_of(A),Nb) = aa($o,char,char2(~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),one_one(nat)),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,one2))))),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_6100_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),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ).

% of_char_mod_256
tff(fact_6101_nat__of__char__less__256,axiom,
    ! [C2: char] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ).

% nat_of_char_less_256
tff(fact_6102_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),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ).

% range_nat_of_char
tff(fact_6103_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A,C2: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),Nb) = C2 )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))),Nb) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ) ) ).

% char_of_eq_iff
tff(fact_6104_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(aa($o,char,char2((B0),(B1),(B22),(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($o,code_integer,zero_neq_one_of_bool(code_integer),(B72))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B62)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B52)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B42)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B32)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B22)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B1)))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B0))) ).

% integer_of_char_code
tff(fact_6105_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aTP_Lamp_vq(code_integer,fun($o,char))),code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_6106_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_6107_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),removeAll(list(A),nil(A),Xs))
        & ! [Ys4: list(A)] :
            ( aa(set(list(A)),$o,member(list(A),Ys4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys4) )
        & ! [Ys4: list(A),Zs3: list(A)] :
            ( ( aa(set(list(A)),$o,member(list(A),Ys4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & aa(set(list(A)),$o,member(list(A),Zs3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & ( Ys4 != Zs3 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys4)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_6108_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_6109_set__empty,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xs) = bot_bot(set(A)) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty
tff(fact_6110_set__empty2,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty2
tff(fact_6111_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_6112_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_6113_list_Osize_I3_J,axiom,
    ! [A: $tType] : aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ).

% list.size(3)
tff(fact_6114_Pow__set_I1_J,axiom,
    ! [A: $tType] : pow2(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),insert(set(A),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_6115_of__char__Char,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : aa(char,A,comm_s6883823935334413003f_char(A),aa($o,char,char2((B0),(B1),(B22),(B32),(B42),(B52),(B62)),(B72))) = groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),cons($o,(B0),cons($o,(B1),cons($o,(B22),cons($o,(B32),cons($o,(B42),cons($o,(B52),cons($o,(B62),cons($o,(B72),nil($o)))))))))) ) ).

% of_char_Char
tff(fact_6116_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D5: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( Xb != zero_zero(real) )
       => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
             => ( D5 = 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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
               => ( D5 = 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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
               => ( D5 = 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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Nb),D5,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_6117_nth__Cons__Suc,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,cons(A,Xb,Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ).

% nth_Cons_Suc
tff(fact_6118_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),cons(A,X21,X222)) = aa(set(A),set(A),insert(A,X21),aa(list(A),set(A),set2(A),X222)) ).

% list.simps(15)
tff(fact_6119_nth__Cons__0,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,A,nth(A,cons(A,Xb,Xs)),zero_zero(nat)) = Xb ).

% nth_Cons_0
tff(fact_6120_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A2: A,Xb: B,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F2,A2,cons(B,Xb,Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F2,Xb)),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_6121_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : enumerate(A,Nb,cons(A,Xb,Xs)) = cons(product_prod(nat,A),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Nb),Xb),enumerate(A,aa(nat,nat,suc,Nb),Xs)) ).

% enumerate_simps(2)
tff(fact_6122_nth__Cons__numeral,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),V: num] : aa(nat,A,nth(A,cons(A,Xb,Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_6123_nth__Cons__pos,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,A,nth(A,cons(A,Xb,Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_6124_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
       => ( ! [X2: A,Xs2: list(A),Y4: B,Ys3: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ys3)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,cons(A,X2,Xs2)),cons(B,Y4,Ys3)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_6125_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs2: list(C),P: fun(list(A),fun(list(B),fun(list(C),$o)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs2) )
       => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X2: A,Xs2: list(A),Y4: B,Ys3: list(B),Z4: C,Zs: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs) )
                 => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs2),Ys3),Zs)
                   => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,cons(A,X2,Xs2)),cons(B,Y4,Ys3)),cons(C,Z4,Zs)) ) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs),Ys),Zs2) ) ) ) ) ).

% list_induct3
tff(fact_6126_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs2: list(C),Ws: list(D),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D),$o))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs2) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs2) = aa(list(D),nat,size_size(list(D)),Ws) )
         => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,nil(A)),nil(B)),nil(C)),nil(D))
           => ( ! [X2: A,Xs2: list(A),Y4: B,Ys3: list(B),Z4: C,Zs: list(C),W2: D,Ws2: list(D)] :
                  ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
                 => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs) )
                   => ( ( aa(list(C),nat,size_size(list(C)),Zs) = aa(list(D),nat,size_size(list(D)),Ws2) )
                     => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,Xs2),Ys3),Zs),Ws2)
                       => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,cons(A,X2,Xs2)),cons(B,Y4,Ys3)),cons(C,Z4,Zs)),cons(D,W2,Ws2)) ) ) ) )
             => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,Xs),Ys),Zs2),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_6127_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Xb: A,Z: A] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_vr(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_6128_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S3: set(A),G: fun(A,A),G5: fun(A,A),F2: fun(A,A),F6: A,Xb: A] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),S3)
             => has_field_derivative(A,G,aa(A,A,G5,X2),topolo174197925503356063within(A,X2,top_top(set(A)))) )
         => ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => ( aa(set(A),$o,member(A,aa(A,A,F2,Xb)),S3)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_vs(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G5,aa(A,A,F2,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_6129_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G5: fun(A,A),F2: fun(A,A),F6: A,Xb: A] :
          ( ! [X2: A] : has_field_derivative(A,G,aa(A,A,G5,X2),topolo174197925503356063within(A,X2,top_top(set(A))))
         => ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_vs(fun(A,A),fun(fun(A,A),fun(A,A)),G),F2),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G5,aa(A,A,F2,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_6130_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xb: A,Db: A,S3: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_vs(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,Xb,S3)) ) ) ) ).

% DERIV_chain2
tff(fact_6131_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,aa(A,A,F2,Xb),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_vt(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),E5),D5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_chain'
tff(fact_6132_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Mb: A,Xb: A] :
          ( has_field_derivative(A,G,Mb,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_vu(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G,Xb))),Mb),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_6133_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Mb: A,Xb: A] :
          ( has_field_derivative(A,G,Mb,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_vv(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,Xb))),Mb),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_6134_DERIV__mirror,axiom,
    ! [F2: fun(real,real),Y: real,Xb: real] :
      ( has_field_derivative(real,F2,Y,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),Xb),top_top(set(real))))
    <=> has_field_derivative(real,aTP_Lamp_vw(fun(real,real),fun(real,real),F2),aa(real,real,uminus_uminus(real),Y),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_mirror
tff(fact_6135_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))),aa(real,real,F2,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_6136_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_6137_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_6138_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4))),aa(real,real,F2,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_6139_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X2: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X2,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(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,aa(real,fun(real,real),minus_minus(real),B2),A2)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_6140_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X2: real] : has_field_derivative(real,F2,K,topolo174197925503356063within(real,X2,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_6141_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,Xb: real,Y: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( aa(set(real),$o,member(real,Xb),set_or5935395276787703475ssThan(real,A2,B2))
       => ( aa(set(real),$o,member(real,Y),set_or5935395276787703475ssThan(real,A2,B2))
         => ( ! [X2: real] :
                ( aa(set(real),$o,member(real,X2),set_or5935395276787703475ssThan(real,A2,B2))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X2,top_top(set(real)))) )
           => ( aa(real,real,F2,Xb) = aa(real,real,F2,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_6142_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_6143_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y5) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_6144_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun(real,real),G5: fun(real,real)] :
      ( ! [X2: real] :
          ( aa(set(real),$o,member(real,X2),set_or1337092689740270186AtMost(real,A2,B2))
         => has_field_derivative(real,G,aa(real,real,G5,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) )
     => ( ! [X2: real] :
            ( aa(set(real),$o,member(real,X2),set_or1337092689740270186AtMost(real,A2,B2))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,G5,X2)) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,G,A2)),aa(real,real,G,B2)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_6145_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_6146_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y5) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_6147_remdups_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      remdups(A,cons(A,Xb,Xs)) = $ite(aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs)),remdups(A,Xs),cons(A,Xb,remdups(A,Xs))) ).

% remdups.simps(2)
tff(fact_6148_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( distinct(A,cons(A,Xb,Xs))
    <=> ( ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
        & distinct(A,Xs) ) ) ).

% distinct.simps(2)
tff(fact_6149_list__update__code_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I2: nat,Y: A] : list_update(A,cons(A,Xb,Xs),aa(nat,nat,suc,I2),Y) = cons(A,Xb,list_update(A,Xs,I2,Y)) ).

% list_update_code(3)
tff(fact_6150_DERIV__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [S: set(A),F2: fun(B,fun(A,B)),F6: fun(C,fun(A,B)),Xb: C,F4: filter(B)] :
          ( ! [N: A] :
              ( aa(set(A),$o,member(A,N),S)
             => has_field_derivative(B,aa(A,fun(B,B),aTP_Lamp_vx(fun(B,fun(A,B)),fun(A,fun(B,B)),F2),N),aa(A,B,aa(C,fun(A,B),F6,Xb),N),F4) )
         => has_field_derivative(B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_vy(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),F6,Xb)),S),F4) ) ) ).

% DERIV_sum
tff(fact_6151_length__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),cons(A,Xb,Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_6152_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [Y3: A,Ys4: list(A)] :
          ( ( Xs = cons(A,Y3,Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_6153_Suc__length__conv,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,Nb) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y3: A,Ys4: list(A)] :
          ( ( Xs = cons(A,Y3,Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_6154_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,F4: filter(A),G: fun(A,A),G5: A] :
          ( has_field_derivative(A,F2,F6,F4)
         => ( has_field_derivative(A,G,G5,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_vz(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),F6),G5),F4) ) ) ) ).

% field_differentiable_diff
tff(fact_6155_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_vz(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),minus_minus(A),D5),E5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_diff
tff(fact_6156_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F4: filter(A)] : has_field_derivative(A,aTP_Lamp_wa(A,A),one_one(A),F4) ) ).

% DERIV_ident
tff(fact_6157_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,F4: filter(A),G: fun(A,A),G5: A] :
          ( has_field_derivative(A,F2,F6,F4)
         => ( has_field_derivative(A,G,G5,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wb(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F6),G5),F4) ) ) ) ).

% field_differentiable_add
tff(fact_6158_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wb(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D5),E5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_add
tff(fact_6159_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,Xb: A,S3: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wc(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,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F2,Xb))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_mult
tff(fact_6160_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wc(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,Xb)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_mult'
tff(fact_6161_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aTP_Lamp_wd(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,Xb))),Db),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivative_cosh
tff(fact_6162_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aTP_Lamp_we(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,Xb))),Db),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivative_sinh
tff(fact_6163_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),C2: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_wf(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),D5),C2),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_cmult_right
tff(fact_6164_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),C2: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_wg(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D5),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_cmult
tff(fact_6165_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,F4: filter(A),C2: real] :
          ( has_field_derivative(A,F2,D5,F4)
         => has_field_derivative(A,aa(real,fun(A,A),aTP_Lamp_wh(fun(A,A),fun(real,fun(A,A)),F2),C2),aa(A,A,real_V8093663219630862766scaleR(A,C2),D5),F4) ) ) ).

% has_field_derivative_scaleR_right
tff(fact_6166_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_wi(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,Xb))),D5)),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xb)))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_inverse'
tff(fact_6167_DERIV__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aTP_Lamp_wj(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),D5),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_minus
tff(fact_6168_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,F6,F4)
         => has_field_derivative(A,aTP_Lamp_wj(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),F6),F4) ) ) ).

% field_differentiable_minus
tff(fact_6169_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F4: filter(A)] : has_field_derivative(A,aTP_Lamp_wk(A,fun(A,A),K),zero_zero(A),F4) ) ).

% DERIV_const
tff(fact_6170_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I2: nat,V: A] : list_update(A,cons(A,Xb,Xs),I2,V) = case_nat(list(A),cons(A,V,Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_wl(A,fun(list(A),fun(A,fun(nat,list(A)))),Xb),Xs),V),I2) ).

% list_update.simps(2)
tff(fact_6171_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(set(real),$o,member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4)),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_6172_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(set(real),$o,member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4)),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))),aa(real,real,F2,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_6173_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(set(real),$o,member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4)),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4))),aa(real,real,F2,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_6174_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( aa(set(real),$o,member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4)),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xb)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H4))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_6175_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,Xb: A,S3: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C2),C2,topolo174197925503356063within(A,Xb,S3)) ) ).

% DERIV_cmult_Id
tff(fact_6176_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( aa(set(A),$o,member(A,Y),aa(list(A),set(A),set2(A),X222))
     => aa(set(A),$o,member(A,Y),aa(list(A),set(A),set2(A),cons(A,X21,X222))) ) ).

% list.set_intros(2)
tff(fact_6177_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(set(A),$o,member(A,X21),aa(list(A),set(A),set2(A),cons(A,X21,X222))) ).

% list.set_intros(1)
tff(fact_6178_list_Oset__cases,axiom,
    ! [A: $tType,E: A,A2: list(A)] :
      ( aa(set(A),$o,member(A,E),aa(list(A),set(A),set2(A),A2))
     => ( ! [Z23: list(A)] : A2 != cons(A,E,Z23)
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( A2 = cons(A,Z12,Z23) )
             => ~ aa(set(A),$o,member(A,E),aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_6179_set__ConsD,axiom,
    ! [A: $tType,Y: A,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Y),aa(list(A),set(A),set2(A),cons(A,Xb,Xs)))
     => ( ( Y = Xb )
        | aa(set(A),$o,member(A,Y),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% set_ConsD
tff(fact_6180_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),G: fun(A,A),E5: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E5,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wm(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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,Xb)),E5))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,Xb)),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% DERIV_divide
tff(fact_6181_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),C2: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_wn(fun(A,A),fun(A,fun(A,A)),F2),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D5),C2),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_cdivide
tff(fact_6182_replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : replicate(A,aa(nat,nat,suc,Nb),Xb) = cons(A,Xb,replicate(A,Nb,Xb)) ).

% replicate_Suc
tff(fact_6183_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))
     => ( Xs != cons(A,Xb,Ys) ) ) ).

% impossible_Cons
tff(fact_6184_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),cons(A,Xb,Xs))) ).

% set_subset_Cons
tff(fact_6185_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xb: A,Db: A,S3: set(A)] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_chain
tff(fact_6186_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Da: A,G: fun(A,A),Xb: A,S3: set(A),Db: A] :
          ( has_field_derivative(A,F2,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),aa(set(A),set(A),image(A,A,G),S3)))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_image_chain
tff(fact_6187_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xb),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S)))
         => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F2),aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_6188_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Y: A,Z: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xb),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_wo(fun(A,A),fun(A,fun(A,A)),F2),Z),Y,topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_at_within_shift
tff(fact_6189_MVT2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F6: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
             => has_field_derivative(real,F2,aa(real,real,F6,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
       => ? [Z4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z4),B2)
            & ( aa(real,real,aa(real,fun(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,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(real,real,F6,Z4)) ) ) ) ) ).

% MVT2
tff(fact_6190_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y4))),D2)
             => ( aa(real,real,F2,Xb) = aa(real,real,F2,Y4) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_6191_DERIV__ln,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_6192_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Mb: A,Xb: A] :
          ( has_field_derivative(A,G,Mb,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_wp(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,Xb)))),Mb),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_6193_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xaa: A] : has_field_derivative(A,aTP_Lamp_wq(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xaa),K))),topolo174197925503356063within(A,Xaa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_6194_nth__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,cons(A,Xb,Xs)),Nb) = case_nat(A,Xb,nth(A,Xs),Nb) ).

% nth_Cons
tff(fact_6195_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_wr(fun(A,A),fun(nat,fun(A,A)),F2),Nb),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),Nb))),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xb)),Nb))),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_power_Suc
tff(fact_6196_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X2: real] : has_field_derivative(real,V,K,topolo174197925503356063within(real,X2,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),aa(num,num,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),aa(num,num,bit0,one2))) ) ) ) ).

% DERIV_const_average
tff(fact_6197_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,S3: set(A)] :
          ( ( Xb != 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),Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_inverse
tff(fact_6198_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S3: set(A),Nb: nat] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_ws(fun(A,A),fun(nat,fun(A,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_power
tff(fact_6199_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y4))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Y4)),aa(real,real,F2,Xb)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_6200_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y4))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Xb)),aa(real,real,F2,Y4)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_6201_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_6202_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S3: set(real)] : has_field_derivative(real,aTP_Lamp_wt(nat,fun(real,real),Nb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,Xb,S3)) ).

% DERIV_pow
tff(fact_6203_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [Y4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),Y4))
         => has_field_derivative(A,aTP_Lamp_wu(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_6204_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo174197925503356063within(A,Xb,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,Xb,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_6205_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),Mb: real,Xb: real,Nb: nat] :
      ( has_field_derivative(real,G,Mb,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_wv(fun(real,real),fun(nat,fun(real,real)),G),Nb),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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,G,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))))),Mb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_6206_Suc__le__length__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X: A,Ys4: list(A)] :
          ( ( Xs = cons(A,X,Ys4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_6207_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( topolo174197925503356063within(A,B2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_6208_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xb: A,S3: set(A),G: fun(A,A),E: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_wm(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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),E),aa(A,A,F2,Xb)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% DERIV_quotient
tff(fact_6209_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xb,S3))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_wi(fun(A,A),fun(A,A),F2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F2,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_6210_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),F2: fun(A,A),F6: A,Z: A] :
          ( ! [Z4: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z4)),K5)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),aa(A,A,F2,Z4)) )
         => ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),F6) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_6211_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] :
      aa(A,nat,count_list(A,cons(A,Xb,Xs)),Y) = $ite(Xb = Y,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y)),one_one(nat)),aa(A,nat,count_list(A,Xs),Y)) ).

% count_list.simps(2)
tff(fact_6212_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_ww(real,fun(real,real),R),aa(real,real,aa(real,fun(real,real),times_times(real),R),powr(real,Z,aa(real,real,aa(real,fun(real,real),minus_minus(real),R),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_6213_the__elem__set,axiom,
    ! [A: $tType,Xb: A] : the_elem(A,aa(list(A),set(A),set2(A),cons(A,Xb,nil(A)))) = Xb ).

% the_elem_set
tff(fact_6214_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),Z: A] :
          ( ! [Z4: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z4)),K5)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
           => has_field_derivative(A,aTP_Lamp_wu(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_6215_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => has_field_derivative(A,aTP_Lamp_wu(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_6216_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_wx(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
               => has_field_derivative(A,aTP_Lamp_wu(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_6217_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),Mb: real,Xb: real,R: real] :
      ( has_field_derivative(real,G,Mb,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,Xb))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_wy(fun(real,real),fun(real,fun(real,real)),G),R),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),powr(real,aa(real,real,G,Xb),aa(real,real,aa(real,fun(real,real),minus_minus(real),R),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),Mb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_6218_DERIV__log,axiom,
    ! [Xb: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => 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)),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_6219_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(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_6220_DERIV__powr,axiom,
    ! [G: fun(real,real),Mb: real,Xb: real,F2: fun(real,real),R: real] :
      ( has_field_derivative(real,G,Mb,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,Xb))
       => ( has_field_derivative(real,F2,R,topolo174197925503356063within(real,Xb,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_wz(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,Xb),aa(real,real,F2,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R),aa(real,real,ln_ln(real),aa(real,real,G,Xb)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),Mb),aa(real,real,F2,Xb))),aa(real,real,G,Xb)))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_6221_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != 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,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_6222_nth__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      aa(nat,A,nth(A,cons(A,Xb,Xs)),Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ).

% nth_Cons'
tff(fact_6223_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => 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,Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_6224_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F6: fun(real,fun(nat,real)),X0: real,A2: real,B2: real,L5: fun(nat,real)] :
      ( ! [N: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_xa(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F2),N),aa(nat,real,aa(real,fun(nat,real),F6,X0),N),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X2: real] :
            ( aa(set(real),$o,member(real,X2),set_or5935395276787703475ssThan(real,A2,B2))
           => summable(real,aa(real,fun(nat,real),F2,X2)) )
       => ( aa(set(real),$o,member(real,X0),set_or5935395276787703475ssThan(real,A2,B2))
         => ( summable(real,aa(real,fun(nat,real),F6,X0))
           => ( summable(real,L5)
             => ( ! [N: nat,X2: real,Y4: real] :
                    ( aa(set(real),$o,member(real,X2),set_or5935395276787703475ssThan(real,A2,B2))
                   => ( aa(set(real),$o,member(real,Y4),set_or5935395276787703475ssThan(real,A2,B2))
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),F2,X2),N)),aa(nat,real,aa(real,fun(nat,real),F2,Y4),N)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L5,N)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X2),Y4)))) ) )
               => has_field_derivative(real,aTP_Lamp_xb(fun(real,fun(nat,real)),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),F6,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_6225_DERIV__arctan,axiom,
    ! [Xb: 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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_6226_arsinh__real__has__field__derivative,axiom,
    ! [Xb: 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),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,Xb,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_6227_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) != 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,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_6228_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X21: A,X222: list(A)] : size_list(A,Xb,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,Xb,X21)),size_list(A,Xb,X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6229_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Xb: A,Db: A,S3: set(A)] :
          ( ( cosh(A,aa(A,A,G,Xb)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aTP_Lamp_xc(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tanh(A),aa(A,A,G,Xb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Db),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_6230_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D5: real] :
      ( ( Xb != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( D5 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
       => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
           => ( D5 = 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,Xb)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
         => has_field_derivative(real,sqrt,D5,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_6231_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => 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,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,Xb,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_6232_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),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,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(real,Xb,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_6233_nth__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      ( ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( aa(nat,A,nth(A,cons(A,Xb,Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6234_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A),Nb: nat] :
      ( ( Xb != Y )
     => ( ( aa(nat,A,nth(A,cons(A,Xb,Xs)),Nb) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6235_Cons__replicate__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat,Y: A] :
      ( ( cons(A,Xb,Xs) = replicate(A,Nb,Y) )
    <=> ( ( Xb = Y )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_6236_DERIV__power__series_H,axiom,
    ! [R2: real,F2: fun(nat,real),X0: real] :
      ( ! [X2: real] :
          ( aa(set(real),$o,member(real,X2),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_xd(fun(nat,real),fun(real,fun(nat,real)),F2),X2)) )
     => ( aa(set(real),$o,member(real,X0),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R2),R2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => has_field_derivative(real,aTP_Lamp_xf(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_xd(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_6237_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
       => has_field_derivative(real,root(Nb),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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_6238_DERIV__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_6239_DERIV__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_6240_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xb: real,Nb: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X2),topolo174197925503356063within(real,X2,top_top(set(real))))
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F2,Xb) = 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_xg(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_6241_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xb: real,Nb: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
        & ! [M: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X2),topolo174197925503356063within(real,X2,top_top(set(real)))) )
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
          & ( aa(real,real,F2,Xb) = 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_xg(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_6242_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
     => ( ( Xb != zero_zero(real) )
       => has_field_derivative(real,root(Nb),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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_6243_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M: nat,T4: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),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_xh(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_6244_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ? [T4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T4)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),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_xh(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_6245_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),zero_zero(real))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M: nat,T4: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),H),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),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_xh(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_6246_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( Xb != zero_zero(real) )
         => ( ! [M: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),X2),topolo174197925503356063within(real,X2,top_top(set(real))))
           => ? [T4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4))
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
                & ( aa(real,real,F2,Xb) = 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_xg(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_6247_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M: nat,T4: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
            & ( aa(real,real,F2,Xb) = 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_xg(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_6248_Taylor,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),B2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xb)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),B2)
                 => ( ( Xb != C2 )
                   => ? [T4: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),C2),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T4)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),C2) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T4)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),Xb) ) )
                        & ( aa(real,real,F2,Xb) = 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_xi(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),Xb)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),C2)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_6249_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),B2)
             => ? [T4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),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_xj(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_6250_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A2: real,B2: real,C2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M: nat,T4: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),B2)
             => ? [T4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),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_xj(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C2)),aa(nat,set(nat),set_ord_lessThan(nat),Nb))),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,Nb),T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_6251_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B4: real] :
      ( ! [M: nat,T4: real] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
     => ( ( Nb = aa(nat,nat,suc,K) )
       => ! [M2: nat,T6: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Nb)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T6)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T6),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_xl(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Nb),Diff),B4),M2),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T6)),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_xm(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T6)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M2))))),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),T6),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M2)))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,M2))))))),topolo174197925503356063within(real,T6,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_6252_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_xn(real,real),suminf(real,aTP_Lamp_xo(real,fun(nat,real),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_6253_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),zero_zero(real))
         => has_field_derivative(real,root(Nb),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),Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_6254_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S3: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
             => has_derivative(A,real,aTP_Lamp_xp(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_xq(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_6255_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S3: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
             => has_derivative(A,real,aTP_Lamp_xr(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_xs(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_6256_has__derivative__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),G: fun(B,C),G5: fun(B,C)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(B,C,G,G5,topolo174197925503356063within(B,aa(A,B,F2,Xb),top_top(set(B))))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xt(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_xt(fun(A,B),fun(fun(B,C),fun(A,C)),F6),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_compose
tff(fact_6257_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),F6: fun(A,real),Xb: A,S3: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,real,F2,F6,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S3))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xu(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_xv(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),Xb),G),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_scaleR
tff(fact_6258_has__field__derivative__imp__has__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,D5,F4)
         => has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D5),F4) ) ) ).

% has_field_derivative_imp_has_derivative
tff(fact_6259_has__derivative__imp__has__field__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: fun(A,A),F4: filter(A),D7: A] :
          ( has_derivative(A,A,F2,D5,F4)
         => ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X2),D7) = aa(A,A,D5,X2)
           => has_field_derivative(A,F2,D7,F4) ) ) ) ).

% has_derivative_imp_has_field_derivative
tff(fact_6260_has__field__derivative__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,F4: filter(A)] :
          ( has_field_derivative(A,F2,D5,F4)
        <=> has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D5),F4) ) ) ).

% has_field_derivative_def
tff(fact_6261_has__derivative__in__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),G: fun(B,C),G5: fun(B,C)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(B,C,G,G5,topolo174197925503356063within(B,aa(A,B,F2,Xb),aa(set(A),set(B),image(A,B,F2),S3)))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xt(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),aa(fun(B,C),fun(A,C),aTP_Lamp_xt(fun(A,B),fun(fun(B,C),fun(A,C)),F6),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_in_compose
tff(fact_6262_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F4: filter(A)] : has_derivative(A,B,aTP_Lamp_xw(B,fun(A,B),C2),aTP_Lamp_xx(A,B),F4) ) ).

% has_derivative_const
tff(fact_6263_has__derivative__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F6,F4)
         => has_derivative(A,B,aTP_Lamp_xy(fun(A,B),fun(A,B),F2),aTP_Lamp_xy(fun(A,B),fun(A,B),F6),F4) ) ) ).

% has_derivative_minus
tff(fact_6264_has__derivative__scaleR__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,B),G5: fun(A,B),F4: filter(A),R: real] :
          ( has_derivative(A,B,G,G5,F4)
         => has_derivative(A,B,aa(real,fun(A,B),aTP_Lamp_xz(fun(A,B),fun(real,fun(A,B)),G),R),aa(real,fun(A,B),aTP_Lamp_xz(fun(A,B),fun(real,fun(A,B)),G5),R),F4) ) ) ).

% has_derivative_scaleR_right
tff(fact_6265_has__derivative__scaleR__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G5: fun(A,real),F4: filter(A),Xb: B] :
          ( has_derivative(A,real,G,G5,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_ya(fun(A,real),fun(B,fun(A,B)),G),Xb),aa(B,fun(A,B),aTP_Lamp_ya(fun(A,real),fun(B,fun(A,B)),G5),Xb),F4) ) ) ).

% has_derivative_scaleR_left
tff(fact_6266_has__derivative__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G5: fun(A,real),F4: filter(A)] :
          ( has_derivative(A,real,G,G5,F4)
         => has_derivative(A,B,aTP_Lamp_yb(fun(A,real),fun(A,B),G),aTP_Lamp_yb(fun(A,real),fun(A,B),G5),F4) ) ) ).

% has_derivative_of_real
tff(fact_6267_has__derivative__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F4: filter(A)] : has_derivative(A,A,aTP_Lamp_yc(A,A),aTP_Lamp_yc(A,A),F4) ) ).

% has_derivative_ident
tff(fact_6268_has__derivative__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G5: fun(A,B),F4: filter(A),Xb: B] :
          ( has_derivative(A,B,G,G5,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_yd(fun(A,B),fun(B,fun(A,B)),G),Xb),aa(B,fun(A,B),aTP_Lamp_yd(fun(A,B),fun(B,fun(A,B)),G5),Xb),F4) ) ) ).

% has_derivative_mult_right
tff(fact_6269_has__derivative__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G5: fun(A,B),F4: filter(A),Y: B] :
          ( has_derivative(A,B,G,G5,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_ye(fun(A,B),fun(B,fun(A,B)),G),Y),aa(B,fun(A,B),aTP_Lamp_ye(fun(A,B),fun(B,fun(A,B)),G5),Y),F4) ) ) ).

% has_derivative_mult_left
tff(fact_6270_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F4: filter(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,F4)
         => ( has_derivative(A,B,G,G5,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_yf(fun(A,B),fun(fun(A,B),fun(A,B)),F6),G5),F4) ) ) ) ).

% has_derivative_add
tff(fact_6271_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F4: filter(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,F4)
         => ( has_derivative(A,B,G,G5,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yg(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aTP_Lamp_yg(fun(A,B),fun(fun(A,B),fun(A,B)),F6),G5),F4) ) ) ) ).

% has_derivative_diff
tff(fact_6272_has__derivative__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F6: fun(A,fun(B,C)),F4: filter(B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I3),aa(A,fun(B,C),F6,I3),F4) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yi(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yi(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F6),F4) ) ) ).

% has_derivative_sum
tff(fact_6273_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S3))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_yk(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),Xb),G),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_mult
tff(fact_6274_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F4: fun(A,B),Xb: A] :
          ( has_derivative(A,B,aTP_Lamp_xx(A,B),F4,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => ! [X4: A] : aa(A,B,F4,X4) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_6275_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Ta: set(A),G: fun(A,B),G5: fun(A,fun(A,B)),F2: fun(C,A),S3: set(C),Xb: C,F6: fun(C,A)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),Ta)
             => has_derivative(A,B,G,aa(A,fun(A,B),G5,X2),topolo174197925503356063within(A,X2,Ta)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F2),S3)),Ta)
           => ( aa(set(C),$o,member(C,Xb),S3)
             => ( has_derivative(C,A,F2,F6,topolo174197925503356063within(C,Xb,S3))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yl(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_ym(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G5),F2),Xb),F6),topolo174197925503356063within(C,Xb,S3)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_6276_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,S3: set(A)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,real,aTP_Lamp_yn(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_yo(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G5),Xb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_exp
tff(fact_6277_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,S3: set(A)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,real,aTP_Lamp_yp(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_yq(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G5),Xb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_sin
tff(fact_6278_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S3: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,A,aTP_Lamp_we(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,Xb))),Db)),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_sinh
tff(fact_6279_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S3: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,A,aTP_Lamp_wd(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,Xb))),Db)),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_cosh
tff(fact_6280_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_ys(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),Xb),G),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_6281_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,S: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_yt(A,fun(A,A),Xb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_inverse'
tff(fact_6282_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),Xb: B,F6: fun(B,A),S: set(B)] :
          ( ( aa(B,A,F2,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F6,topolo174197925503356063within(B,Xb,S))
           => has_derivative(B,A,aTP_Lamp_yu(fun(B,A),fun(B,A),F2),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_yv(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),Xb),F6),topolo174197925503356063within(B,Xb,S)) ) ) ) ).

% has_derivative_inverse
tff(fact_6283_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(real,real),F6: real,G: fun(A,real),Xb: A,G5: fun(A,real),S3: set(A)] :
          ( has_field_derivative(real,F2,F6,topolo174197925503356063within(real,aa(A,real,G,Xb),top_top(set(real))))
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_yw(fun(real,real),fun(fun(A,real),fun(A,real)),F2),G),aa(fun(A,real),fun(A,real),aTP_Lamp_yx(real,fun(fun(A,real),fun(A,real)),F6),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_6284_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,S3: set(A)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,real,aTP_Lamp_yy(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_yz(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G5),Xb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_cos
tff(fact_6285_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S: set(A),Nb: nat] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_za(fun(A,B),fun(nat,fun(A,B)),F2),Nb),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_zb(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F2),F6),Xb),Nb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_power
tff(fact_6286_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S3: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
           => has_derivative(A,real,aTP_Lamp_zc(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_zd(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_ln
tff(fact_6287_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ze(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_zf(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F6),Xb),G),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_divide
tff(fact_6288_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F6: fun(A,fun(B,C)),Xb: B,S: set(B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I3),aa(A,fun(B,C),F6,I3),topolo174197925503356063within(B,Xb,S)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_zh(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_zj(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I5),F2),F6),Xb),topolo174197925503356063within(B,Xb,S)) ) ) ).

% has_derivative_prod
tff(fact_6289_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,X5: set(A),F2: fun(A,real),F6: fun(A,real)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,X5))
         => ( has_derivative(A,real,F2,F6,topolo174197925503356063within(A,Xb,X5))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
             => ( aa(set(A),$o,member(A,Xb),X5)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zk(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_zl(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G5),Xb),F2),F6),topolo174197925503356063within(A,Xb,X5)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_6290_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S3: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
           => has_derivative(A,real,aTP_Lamp_zm(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_zn(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_6291_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,S3: set(A)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,real,aTP_Lamp_zo(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_zp(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G5),Xb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_arctan
tff(fact_6292_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S3: set(A)] :
          ( ( cos(real,aa(A,real,G,Xb)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S3))
           => has_derivative(A,real,aTP_Lamp_zq(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_zr(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_tan
tff(fact_6293_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,append(A,cons(A,Xb,nil(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% concat_inth
tff(fact_6294_has__derivative__floor,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [G: fun(B,real),Xb: B,F2: fun(real,A),G5: fun(B,real),S3: set(B)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,aa(B,real,G,Xb),top_top(set(real))),F2)
         => ( ~ aa(set(A),$o,member(A,aa(real,A,F2,aa(B,real,G,Xb))),ring_1_Ints(A))
           => ( has_derivative(B,real,G,G5,topolo174197925503356063within(B,Xb,S3))
             => has_derivative(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_zs(fun(B,real),fun(fun(real,A),fun(B,real)),G),F2),aTP_Lamp_zt(fun(B,real),fun(B,real),G5),topolo174197925503356063within(B,Xb,S3)) ) ) ) ) ).

% has_derivative_floor
tff(fact_6295_append__eq__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Us: list(A),Vs: list(A)] :
      ( ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        | ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
     => ( ( append(A,Xs,Us) = append(A,Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_6296_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_6297_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_6298_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,cons(A,Xb,Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% nth_append_length
tff(fact_6299_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: 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)),Nb)) = aa(nat,A,nth(A,Ys),Nb) ).

% nth_append_length_plus
tff(fact_6300_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A] : list_update(A,append(A,Xs,cons(A,Xb,Ys)),aa(list(A),nat,size_size(list(A)),Xs),Y) = append(A,Xs,cons(A,Y,Ys)) ).

% list_update_length
tff(fact_6301_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,append(A,Xs,Ys))
    <=> ( distinct(A,Xs)
        & distinct(A,Ys)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_6302_isCont__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_zu(fun(A,product_prod(B,C)),fun(A,C),F2)) ) ) ).

% isCont_snd
tff(fact_6303_isCont__fst,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_zv(fun(A,product_prod(B,C)),fun(A,B),F2)) ) ) ).

% isCont_fst
tff(fact_6304_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_zw(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% isCont_Pair
tff(fact_6305_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_zx(fun(A,B),fun(A,real),F2)) ) ) ).

% isCont_norm
tff(fact_6306_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_zy(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% isCont_o2
tff(fact_6307_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),Xb: C,G: fun(A,B),S3: set(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(C,A,F2,Xb),top_top(set(A))),G)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,Xb,S3),F2)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,Xb,S3),aa(fun(A,B),fun(C,B),aTP_Lamp_zz(fun(C,A),fun(fun(A,B),fun(C,B)),F2),G)) ) ) ) ).

% continuous_within_compose3
tff(fact_6308_isCont__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aaa(fun(A,real),fun(A,B),G)) ) ) ).

% isCont_of_real
tff(fact_6309_isCont__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,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_aab(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_scaleR
tff(fact_6310_continuous__ident,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,S: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_aac(A,A)) ) ).

% continuous_ident
tff(fact_6311_split__list,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ? [Ys3: list(A),Zs: list(A)] : Xs = append(A,Ys3,cons(A,Xb,Zs)) ) ).

% split_list
tff(fact_6312_split__list__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ? [Ys3: list(A),Zs: list(A)] :
          ( ( Xs = append(A,Ys3,cons(A,Xb,Zs)) )
          & ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Zs)) ) ) ).

% split_list_last
tff(fact_6313_split__list__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ? [Ys3: list(A),X2: A] :
          ( ? [Zs: list(A)] : Xs = append(A,Ys3,cons(A,X2,Zs))
          & aa(A,$o,P,X2) ) ) ).

% split_list_prop
tff(fact_6314_split__list__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ? [Ys3: list(A),Zs: list(A)] :
          ( ( Xs = append(A,Ys3,cons(A,Xb,Zs)) )
          & ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Ys3)) ) ) ).

% split_list_first
tff(fact_6315_split__list__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ~ ! [Ys3: list(A),X2: A] :
            ( ? [Zs: list(A)] : Xs = append(A,Ys3,cons(A,X2,Zs))
           => ~ aa(A,$o,P,X2) ) ) ).

% split_list_propE
tff(fact_6316_append__Cons__eq__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),Xs4: list(A),Ys5: list(A)] :
      ( ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Ys))
       => ( ( append(A,Xs,cons(A,Xb,Ys)) = append(A,Xs4,cons(A,Xb,Ys5)) )
        <=> ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_6317_in__set__conv__decomp,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys4: list(A),Zs3: list(A)] : Xs = append(A,Ys4,cons(A,Xb,Zs3)) ) ).

% in_set_conv_decomp
tff(fact_6318_split__list__last__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ? [Ys3: list(A),X2: A,Zs: list(A)] :
          ( ( Xs = append(A,Ys3,cons(A,X2,Zs)) )
          & aa(A,$o,P,X2)
          & ! [Xa: A] :
              ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Zs))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_last_prop
tff(fact_6319_split__list__first__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ? [Ys3: list(A),X2: A] :
          ( ? [Zs: list(A)] : Xs = append(A,Ys3,cons(A,X2,Zs))
          & aa(A,$o,P,X2)
          & ! [Xa: A] :
              ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Ys3))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_first_prop
tff(fact_6320_split__list__last__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ~ ! [Ys3: list(A),X2: A,Zs: list(A)] :
            ( ( Xs = append(A,Ys3,cons(A,X2,Zs)) )
           => ( aa(A,$o,P,X2)
             => ~ ! [Xa: A] :
                    ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Zs))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_last_propE
tff(fact_6321_split__list__first__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X4: A] :
          ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X4) )
     => ~ ! [Ys3: list(A),X2: A] :
            ( ? [Zs: list(A)] : Xs = append(A,Ys3,cons(A,X2,Zs))
           => ( aa(A,$o,P,X2)
             => ~ ! [Xa: A] :
                    ( aa(set(A),$o,member(A,Xa),aa(list(A),set(A),set2(A),Ys3))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_first_propE
tff(fact_6322_in__set__conv__decomp__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xs = append(A,Ys4,cons(A,Xb,Zs3)) )
          & ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Zs3)) ) ) ).

% in_set_conv_decomp_last
tff(fact_6323_in__set__conv__decomp__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xs = append(A,Ys4,cons(A,Xb,Zs3)) )
          & ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Ys4)) ) ) ).

% in_set_conv_decomp_first
tff(fact_6324_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
    <=> ? [Ys4: list(A),X: A,Zs3: list(A)] :
          ( ( Xs = append(A,Ys4,cons(A,X,Zs3)) )
          & aa(A,$o,P,X)
          & ! [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),aa(list(A),set(A),set2(A),Zs3))
             => ~ aa(A,$o,P,Xa2) ) ) ) ).

% split_list_last_prop_iff
tff(fact_6325_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
    <=> ? [Ys4: list(A),X: A] :
          ( ? [Zs3: list(A)] : Xs = append(A,Ys4,cons(A,X,Zs3))
          & aa(A,$o,P,X)
          & ! [Xa2: A] :
              ( aa(set(A),$o,member(A,Xa2),aa(list(A),set(A),set2(A),Ys4))
             => ~ aa(A,$o,P,Xa2) ) ) ) ).

% split_list_first_prop_iff
tff(fact_6326_continuous__within__compose2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [Xb: A,S3: set(A),F2: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,S3),F2)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,F2,Xb),aa(set(A),set(B),image(A,B,F2),S3)),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,Xb,S3),aa(fun(B,C),fun(A,C),aTP_Lamp_zy(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G)) ) ) ) ).

% continuous_within_compose2
tff(fact_6327_has__derivative__cnj,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G5: fun(A,complex),F4: filter(A)] :
          ( has_derivative(A,complex,G,G5,F4)
         => has_derivative(A,complex,aTP_Lamp_aad(fun(A,complex),fun(A,complex),G),aTP_Lamp_aad(fun(A,complex),fun(A,complex),G5),F4) ) ) ).

% has_derivative_cnj
tff(fact_6328_has__derivative__Im,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G5: fun(A,complex),F4: filter(A)] :
          ( has_derivative(A,complex,G,G5,F4)
         => has_derivative(A,real,aTP_Lamp_aae(fun(A,complex),fun(A,real),G),aTP_Lamp_aae(fun(A,complex),fun(A,real),G5),F4) ) ) ).

% has_derivative_Im
tff(fact_6329_has__derivative__Re,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,complex),G5: fun(A,complex),F4: filter(A)] :
          ( has_derivative(A,complex,G,G5,F4)
         => has_derivative(A,real,aTP_Lamp_aaf(fun(A,complex),fun(A,real),G),aTP_Lamp_aaf(fun(A,complex),fun(A,real),G5),F4) ) ) ).

% has_derivative_Re
tff(fact_6330_continuous__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),F4: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),S)
             => topolo3448309680560233919inuous(B,C,F4,aa(A,fun(B,C),F2,I3)) )
         => topolo3448309680560233919inuous(B,C,F4,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aah(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2)) ) ) ).

% continuous_prod
tff(fact_6331_continuous__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),F4: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => topolo3448309680560233919inuous(B,C,F4,aa(A,fun(B,C),F2,I3)) )
         => topolo3448309680560233919inuous(B,C,F4,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aaj(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_prod'
tff(fact_6332_remove1__append,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] :
      remove1(A,Xb,append(A,Xs,Ys)) = $ite(aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs)),append(A,remove1(A,Xb,Xs),Ys),append(A,Xs,remove1(A,Xb,Ys))) ).

% remove1_append
tff(fact_6333_continuous__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),F4: filter(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => topolo3448309680560233919inuous(B,C,F4,aa(A,fun(B,C),F2,I3)) )
         => topolo3448309680560233919inuous(B,C,F4,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aal(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_sum
tff(fact_6334_replicate__add,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xb: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb),Xb) = append(A,replicate(A,Nb,Xb),replicate(A,Mb,Xb)) ).

% replicate_add
tff(fact_6335_continuous__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F4: filter(A),F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),F4,F2)
         => topolo3448309680560233919inuous(A,C,F4,aTP_Lamp_zu(fun(A,product_prod(B,C)),fun(A,C),F2)) ) ) ).

% continuous_snd
tff(fact_6336_continuous__fst,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F4: filter(A),F2: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_zv(fun(A,product_prod(B,C)),fun(A,B),F2)) ) ) ).

% continuous_fst
tff(fact_6337_enumerate__append__eq,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : enumerate(A,Nb,append(A,Xs,Ys)) = append(product_prod(nat,A),enumerate(A,Nb,Xs),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_6338_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,C,F4,G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F4,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_zw(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_Pair
tff(fact_6339_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_aam(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_diff
tff(fact_6340_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_aan(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_add
tff(fact_6341_continuous__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_aao(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult
tff(fact_6342_continuous__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_aap(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_mult'
tff(fact_6343_continuous__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aa(B,fun(A,B),aTP_Lamp_aaq(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_left
tff(fact_6344_continuous__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aa(B,fun(A,B),aTP_Lamp_aar(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_mult_right
tff(fact_6345_continuous__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_aas(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_exp
tff(fact_6346_continuous__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_zx(fun(A,B),fun(A,real),F2)) ) ) ).

% continuous_norm
tff(fact_6347_continuous__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,nat,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,nat),fun(A,B),aTP_Lamp_aat(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_power'
tff(fact_6348_continuous__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),C2: B] : topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_aau(B,fun(A,B),C2)) ) ).

% continuous_const
tff(fact_6349_continuous__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,G)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_aaa(fun(A,real),fun(A,B),G)) ) ) ).

% continuous_of_real
tff(fact_6350_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),Nb: nat] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aa(nat,fun(A,B),aTP_Lamp_aav(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% continuous_power
tff(fact_6351_continuous__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_aaw(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sin
tff(fact_6352_continuous__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_aax(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cos
tff(fact_6353_continuous__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_aab(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_scaleR
tff(fact_6354_continuous__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_aay(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_cosh
tff(fact_6355_continuous__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_aaz(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_sinh
tff(fact_6356_continuous__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_aba(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_max
tff(fact_6357_continuous__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_abb(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_minus
tff(fact_6358_IVT2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),B2: B,Y: A,A2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,B2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,A2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( ! [X2: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X2,top_top(set(B))),F2) )
               => ? [X2: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),B2)
                    & ( aa(B,A,F2,X2) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_6359_IVT,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),A2: B,Y: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,A2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,B2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( ! [X2: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X2,top_top(set(B))),F2) )
               => ? [X2: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),B2)
                    & ( aa(B,A,F2,X2) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_6360_isCont__Lb__Ub,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X2: real] :
            ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
       => ? [L6: real,M7: real] :
            ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),aa(real,real,F2,X4))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,X4)),M7) ) )
            & ! [Y5: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),Y5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),M7) )
               => ? [X2: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
                    & ( aa(real,real,F2,X2) = Y5 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_6361_same__length__different,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != Ys )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => ? [Pre: list(A),X2: A,Xs5: list(A),Y4: A,Ys6: list(A)] :
            ( ( X2 != Y4 )
            & ( Xs = append(A,Pre,append(A,cons(A,X2,nil(A)),Xs5)) )
            & ( Ys = append(A,Pre,append(A,cons(A,Y4,nil(A)),Ys6)) ) ) ) ) ).

% same_length_different
tff(fact_6362_isCont__real__sqrt,axiom,
    ! [Xb: real] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),sqrt) ).

% isCont_real_sqrt
tff(fact_6363_isCont__real__root,axiom,
    ! [Xb: real,Nb: nat] : topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),root(Nb)) ).

% isCont_real_root
tff(fact_6364_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_abc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_6365_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_aao(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_mult
tff(fact_6366_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_aan(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_add
tff(fact_6367_not__distinct__conv__prefix,axiom,
    ! [A: $tType,As: list(A)] :
      ( ~ distinct(A,As)
    <=> ? [Xs3: list(A),Y3: A,Ys4: list(A)] :
          ( aa(set(A),$o,member(A,Y3),aa(list(A),set(A),set2(A),Xs3))
          & distinct(A,Xs3)
          & ( As = append(A,Xs3,cons(A,Y3,Ys4)) ) ) ) ).

% not_distinct_conv_prefix
tff(fact_6368_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_abd(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% isCont_diff
tff(fact_6369_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_abe(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_minus
tff(fact_6370_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),Nb: 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_aav(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% isCont_power
tff(fact_6371_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_abf(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_6372_isCont__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [A3: set(A),A2: B,F2: fun(A,fun(B,C))] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A2,top_top(set(B))),aa(A,fun(B,C),F2,X2)) )
         => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A2,top_top(set(B))),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aal(set(A),fun(fun(A,fun(B,C)),fun(B,C)),A3),F2)) ) ) ).

% isCont_sum
tff(fact_6373_list__update__append1,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Ys: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,append(A,Xs,Ys),I2,Xb) = append(A,list_update(A,Xs,I2,Xb),Ys) ) ) ).

% list_update_append1
tff(fact_6374_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_abg(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_6375_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_aax(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_cos'
tff(fact_6376_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_aaw(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_sin'
tff(fact_6377_isCont__exp_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_aas(fun(A,B),fun(A,B),F2)) ) ) ).

% isCont_exp'
tff(fact_6378_remove1__split,axiom,
    ! [A: $tType,A2: A,Xs: list(A),Ys: list(A)] :
      ( aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),Xs))
     => ( ( remove1(A,A2,Xs) = Ys )
      <=> ? [Ls: list(A),Rs: list(A)] :
            ( ( Xs = append(A,Ls,cons(A,A2,Rs)) )
            & ~ aa(set(A),$o,member(A,A2),aa(list(A),set(A),set2(A),Ls))
            & ( Ys = append(A,Ls,Rs) ) ) ) ) ).

% remove1_split
tff(fact_6379_isCont__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Z: A,Nb: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z,top_top(set(A))),aTP_Lamp_abh(nat,fun(A,A),Nb)) ) ).

% isCont_pochhammer
tff(fact_6380_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
           => ? [M7: A] :
              ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X4)),M7) ) ) ) ) ).

% isCont_bounded
tff(fact_6381_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
           => ? [M7: A] :
                ( ! [X4: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X4)),M7) )
                & ? [X2: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
                    & ( aa(real,A,F2,X2) = M7 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_6382_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
           => ? [M7: A] :
                ( ! [X4: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M7),aa(real,A,F2,X4)) )
                & ? [X2: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
                    & ( aa(real,A,F2,X2) = M7 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_6383_isCont__inverse__function2,axiom,
    ! [A2: real,Xb: real,B2: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
       => ( ! [Z4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z4),B2)
               => ( aa(real,real,G,aa(real,real,F2,Z4)) = Z4 ) ) )
         => ( ! [Z4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z4),B2)
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z4,top_top(set(real))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xb),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_6384_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_abc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_6385_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,cons(A,Xb,nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_6386_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [Y3: A,Ys4: list(A)] :
          ( ( Xs = append(A,Ys4,cons(A,Y3,nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% length_Suc_conv_rev
tff(fact_6387_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] :
      aa(nat,A,nth(A,append(A,Xs,Ys)),Nb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,A,nth(A,Xs),Nb),aa(nat,A,nth(A,Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)))) ).

% nth_append
tff(fact_6388_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_abg(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_6389_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( append(A,Xs,Ys) = append(A,Ys,Xs) )
         => ? [N: nat,Zs: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N)
              & ( concat(A,replicate(list(A),N,Zs)) = append(A,Xs,Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_6390_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat,Xb: A] :
      list_update(A,append(A,Xs,Ys),Nb,Xb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),append(A,list_update(A,Xs,Nb,Xb),Ys),append(A,Xs,list_update(A,Ys,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Xb))) ).

% list_update_append
tff(fact_6391_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S3: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S3),F2)
         => ( ( cos(A,aa(A,A,F2,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S3),aTP_Lamp_abi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_6392_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S3: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S3),F2)
         => ( ( sin(A,aa(A,A,F2,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S3),aTP_Lamp_abj(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_6393_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Xb: A,A3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A3),F2)
         => ( ( cosh(B,aa(A,B,F2,Xb)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A3),aTP_Lamp_abk(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_6394_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A,Xb: A] :
          ( has_field_derivative(A,F2,L,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ? [G6: fun(A,A)] :
              ( ! [Z3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,F2,Z3)),aa(A,A,F2,Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G6,Z3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z3),Xb))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),G6)
              & ( aa(A,A,G6,Xb) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_6395_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
           => ? [M7: A] :
                ( ! [X4: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X4)),M7) )
                & ! [N6: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N6),M7)
                   => ? [X2: real] :
                        ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
                        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),N6),aa(real,A,F2,X2)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_6396_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_abi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_6397_isCont__arcosh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_6398_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_6399_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_abj(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_6400_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D5: real,G: fun(real,real),Xb: real,A2: real,B2: real] :
      ( has_field_derivative(real,F2,D5,topolo174197925503356063within(real,aa(real,real,G,Xb),top_top(set(real))))
     => ( ( D5 != zero_zero(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),B2)
           => ( ! [Y4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Y4)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),B2)
                   => ( aa(real,real,F2,aa(real,real,G,Y4)) = Y4 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D5),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_6401_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A,C2: fun(nat,A),Nb: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_abl(fun(nat,A),fun(nat,fun(A,A)),C2),Nb)) ) ).

% isCont_polynom
tff(fact_6402_isCont__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_6403_isCont__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_6404_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [Y4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),Y4))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_wu(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_6405_LIM__less__bound,axiom,
    ! [B2: real,Xb: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B2),Xb)
     => ( ! [X2: real] :
            ( aa(set(real),$o,member(real,X2),set_or5935395276787703475ssThan(real,B2,Xb))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X2)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_6406_isCont__artanh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_6407_isCont__inverse__function,axiom,
    ! [D2: real,Xb: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
     => ( ! [Z4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z4),Xb))),D2)
           => ( aa(real,real,G,aa(real,real,F2,Z4)) = Z4 ) )
       => ( ! [Z4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Z4),Xb))),D2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z4,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xb),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_6408_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real),G5: fun(real,real),F6: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [Z4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z4),B2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z4,top_top(set(real))),F2) ) )
       => ( ! [Z4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z4),B2)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z4,top_top(set(real))),G) ) )
         => ( ! [Z4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z4),B2)
                 => has_field_derivative(real,G,aa(real,real,G5,Z4),topolo174197925503356063within(real,Z4,top_top(set(real)))) ) )
           => ( ! [Z4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z4)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z4),B2)
                   => has_field_derivative(real,F2,aa(real,real,F6,Z4),topolo174197925503356063within(real,Z4,top_top(set(real)))) ) )
             => ? [C3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2))),aa(real,real,G5,C3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A2))),aa(real,real,F6,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_6409_floor__has__real__derivative,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Xb: real,F2: fun(real,A)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,Xb,top_top(set(real))),F2)
         => ( ~ aa(set(A),$o,member(A,aa(real,A,F2,Xb)),ring_1_Ints(A))
           => has_field_derivative(real,aTP_Lamp_abm(fun(real,A),fun(real,real),F2),zero_zero(real),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_6410_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_wu(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_6411_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A2: A,F2: fun(A,B),C2: fun(nat,B),K5: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_abn(fun(nat,B),fun(B,fun(nat,B)),C2),K5))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,A2))),real_V7770717601297561774m_norm(B,K5))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_abp(fun(A,B),fun(fun(nat,B),fun(A,B)),F2),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_6412_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A3: set(A)] : set_Cons(A,A3,aa(set(list(A)),set(list(A)),insert(list(A),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image(A,list(A),aTP_Lamp_abq(A,list(A))),A3) ).

% set_Cons_sing_Nil
tff(fact_6413_upto__aux__rec,axiom,
    ! [I2: int,J: int,Js: list(int)] :
      upto_aux(I2,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I2),Js,upto_aux(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),cons(int,J,Js))) ).

% upto_aux_rec
tff(fact_6414_isCont__Re,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_abr(fun(A,complex),fun(A,real),G)) ) ) ).

% isCont_Re
tff(fact_6415_isCont__Im,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_abs(fun(A,complex),fun(A,real),G)) ) ) ).

% isCont_Im
tff(fact_6416_continuous__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_abt(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_rabs
tff(fact_6417_continuous__Re,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F4,G)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_abr(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_Re
tff(fact_6418_continuous__arsinh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_abu(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arsinh
tff(fact_6419_continuous__arctan,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_abv(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_arctan
tff(fact_6420_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_abw(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_real_sqrt
tff(fact_6421_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),Nb: nat] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => topolo3448309680560233919inuous(A,real,F4,aa(nat,fun(A,real),aTP_Lamp_abx(fun(A,real),fun(nat,fun(A,real)),F2),Nb)) ) ) ).

% continuous_real_root
tff(fact_6422_continuous__Im,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F4,G)
         => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_abs(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_Im
tff(fact_6423_continuous__complex__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F4,F2)
        <=> ( topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_abr(fun(A,complex),fun(A,real),F2))
            & topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_abs(fun(A,complex),fun(A,real),F2)) ) ) ) ).

% continuous_complex_iff
tff(fact_6424_continuous__cnj,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,F4,G)
         => topolo3448309680560233919inuous(A,complex,F4,aTP_Lamp_aby(fun(A,complex),fun(A,complex),G)) ) ) ).

% continuous_cnj
tff(fact_6425_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_abt(fun(A,real),fun(A,real),F2)) ) ) ).

% isCont_rabs
tff(fact_6426_isCont__cnj,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,G: fun(A,complex)] :
          ( topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A2,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,complex,topolo174197925503356063within(A,A2,top_top(set(A))),aTP_Lamp_aby(fun(A,complex),fun(A,complex),G)) ) ) ).

% isCont_cnj
tff(fact_6427_continuous__at__within__powr,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)
           => ( ( aa(A,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S3),aa(fun(A,real),fun(A,real),aTP_Lamp_abz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_6428_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,S3: set(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xb,S3),F2)
         => ( ( aa(A,real,F2,Xb) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xb,S3),aTP_Lamp_aca(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_within_ln
tff(fact_6429_isCont__powr,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)
           => ( ( aa(A,real,F2,A2) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_abz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% isCont_powr
tff(fact_6430_isCont__ln_H,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xb,top_top(set(A))),F2)
         => ( ( aa(A,real,F2,Xb) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_aca(fun(A,real),fun(A,real),F2)) ) ) ) ).

% isCont_ln'
tff(fact_6431_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S3),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S3),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A2))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),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_acb(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_6432_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A2))
             => ( ( aa(A,real,F2,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),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_acb(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_6433_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = 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),Xb),Xaa))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa),cons(int,Xb,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int)),Xaa)),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),Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_6434_upto_Opsimps,axiom,
    ! [I2: 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),I2),J))
     => ( upto(I2,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J),cons(int,I2,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_6435_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(I2,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_6436_length__upto,axiom,
    ! [I2: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I2,J)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),I2)),one_one(int))) ).

% length_upto
tff(fact_6437_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa(num,int,numeral_numeral(int),Mb),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Mb)),aa(num,int,numeral_numeral(int),Nb)),cons(int,aa(num,int,numeral_numeral(int),Mb),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),Mb)),one_one(int)),aa(num,int,numeral_numeral(int),Nb))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_6438_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_6439_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb)),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),aa(num,int,numeral_numeral(int),Nb)),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Mb))),one_one(int)),aa(num,int,numeral_numeral(int),Nb))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_6440_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa(num,int,numeral_numeral(int),Mb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))),cons(int,aa(num,int,numeral_numeral(int),Mb),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),Mb)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_6441_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I2,K) = append(int,upto(I2,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_6442_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I2,K) = append(int,upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_6443_atLeastLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or7035219750837199246ssThan(int,I2,J) = aa(list(int),set(int),set2(int),upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6444_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( upto(I2,J) = cons(int,I2,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_6445_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa),cons(int,Xb,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int)),Xaa)),nil(int)) ) ) ).

% upto.elims
tff(fact_6446_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      upto(I2,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J),cons(int,I2,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_6447_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( upto(I2,J) = append(int,upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),cons(int,J,nil(int))) ) ) ).

% upto_rec2
tff(fact_6448_greaterThanLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or5935395276787703475ssThan(int,I2,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_6449_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I2,K) = append(int,upto(I2,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(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_6450_GMVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),G: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X2: real] :
            ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X2,top_top(set(real))),F2) )
       => ( ! [X2: real] :
              ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X2,top_top(set(real)))) )
         => ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X2,top_top(set(real))),G) )
           => ( ! [X2: real] :
                  ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X2,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))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),B2)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(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,aa(real,fun(real,real),minus_minus(real),aa(real,real,G,B2)),aa(real,real,G,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_6451_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_wx(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_acd(fun(nat,A),fun(A,fun(A,A)),C2),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_6452_tendsto__const,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [K: B,F4: filter(A)] : filterlim(A,B,aTP_Lamp_ace(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,K),F4) ) ).

% tendsto_const
tff(fact_6453_tendsto__ident__at,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S3: set(A)] : filterlim(A,A,aTP_Lamp_acf(A,A),topolo7230453075368039082e_nhds(A,A2),topolo174197925503356063within(A,A2,S3)) ) ).

% tendsto_ident_at
tff(fact_6454_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_acg(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)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_6455_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,F2: fun(B,A),L: A,F4: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ach(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)),F4)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_6456_power__tendsto__0__iff,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aci(nat,fun(fun(A,real),fun(A,real)),Nb),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_6457_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q2: fun(A,B),C2: B,Ta: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_acj(fun(A,B),fun(B,fun(A,B)),Q2),C2),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_6458_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C2: B,Q2: fun(A,B),Ta: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ack(B,fun(fun(A,B),fun(A,B)),C2),Q2),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C2 = zero_zero(B) )
            | differentiable(A,B,Q2,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_6459_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),F4: filter(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,L,top_top(set(A))),G)
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acl(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F4) ) ) ) ).

% isCont_tendsto_compose
tff(fact_6460_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xb: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acm(A,fun(fun(A,B),fun(A,B)),Xb),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,Xb)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_6461_filterlim__inf,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F32: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F23),F32),F12)
    <=> ( filterlim(A,B,F2,F23,F12)
        & filterlim(A,B,F2,F32,F12) ) ) ).

% filterlim_inf
tff(fact_6462_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F32: filter(B),F23: filter(A),F2: fun(C,A),F12: filter(C)] :
      ( filterlim(A,B,G,F32,F23)
     => ( filterlim(C,A,F2,F23,F12)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acn(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),F32,F12) ) ) ).

% filterlim_compose
tff(fact_6463_filterlim__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filterlim(A,A,aTP_Lamp_ml(A,A),F4,F4) ).

% filterlim_ident
tff(fact_6464_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
         => ( ( A2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aco(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_6465_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_acp(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_6466_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acq(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_6467_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_acr(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_6468_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G: fun(A,B),F4: filter(A),A2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acs(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_6469_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_act(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_6470_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acs(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform2
tff(fact_6471_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),A2: B,F4: filter(A),F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acu(fun(A,B),fun(fun(A,B),fun(A,B)),G),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform
tff(fact_6472_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_act(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_6473_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_act(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_6474_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_6475_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_acw(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_6476_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_acx(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_6477_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acy(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_6478_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
       => filterlim(A,real,aTP_Lamp_acz(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ).

% tendsto_arcosh
tff(fact_6479_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_abj(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F4) ) ) ) ).

% tendsto_cot
tff(fact_6480_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( cosh(B,A2) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_ada(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A2)),F4) ) ) ) ).

% tendsto_tanh
tff(fact_6481_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( cos(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_abi(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F4) ) ) ) ).

% tendsto_tan
tff(fact_6482_tendsto__ln,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( ( 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)),F4) ) ) ).

% tendsto_ln
tff(fact_6483_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_adb(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_6484_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_adb(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_iff
tff(fact_6485_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,real,aTP_Lamp_adb(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_norm_zero
tff(fact_6486_tendsto__powr,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( ( A2 != zero_zero(real) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ).

% tendsto_powr
tff(fact_6487_tendsto__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,B,aTP_Lamp_add(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A2)),F4) ) ) ).

% tendsto_minus
tff(fact_6488_tendsto__minus__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_add(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A2)),F4)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ).

% tendsto_minus_cancel
tff(fact_6489_tendsto__minus__cancel__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Y)),F4)
        <=> filterlim(A,B,aTP_Lamp_add(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,Y),F4) ) ) ).

% tendsto_minus_cancel_left
tff(fact_6490_differentiable__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( differentiable(A,B,F2,F4)
         => differentiable(A,B,aTP_Lamp_xy(fun(A,B),fun(A,B),F2),F4) ) ) ).

% differentiable_minus
tff(fact_6491_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_ade(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A2)),F4) ) ) ) ).

% tendsto_inverse
tff(fact_6492_lim__cnj,axiom,
    ! [A: $tType,F2: fun(A,complex),L: complex,F4: filter(A)] :
      ( filterlim(A,complex,aTP_Lamp_oy(fun(A,complex),fun(A,complex),F2),topolo7230453075368039082e_nhds(complex,cnj(L)),F4)
    <=> filterlim(A,complex,F2,topolo7230453075368039082e_nhds(complex,L),F4) ) ).

% lim_cnj
tff(fact_6493_tendsto__cnj,axiom,
    ! [A: $tType,G: fun(A,complex),A2: complex,F4: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,A2),F4)
     => filterlim(A,complex,aTP_Lamp_oy(fun(A,complex),fun(A,complex),G),topolo7230453075368039082e_nhds(complex,cnj(A2)),F4) ) ).

% tendsto_cnj
tff(fact_6494_tendsto__complex__iff,axiom,
    ! [A: $tType,F2: fun(A,complex),Xb: complex,F4: filter(A)] :
      ( filterlim(A,complex,F2,topolo7230453075368039082e_nhds(complex,Xb),F4)
    <=> ( filterlim(A,real,aTP_Lamp_ot(fun(A,complex),fun(A,real),F2),topolo7230453075368039082e_nhds(real,re(Xb)),F4)
        & filterlim(A,real,aTP_Lamp_ov(fun(A,complex),fun(A,real),F2),topolo7230453075368039082e_nhds(real,im(Xb)),F4) ) ) ).

% tendsto_complex_iff
tff(fact_6495_tendsto__Im,axiom,
    ! [A: $tType,G: fun(A,complex),A2: complex,F4: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,A2),F4)
     => filterlim(A,real,aTP_Lamp_ov(fun(A,complex),fun(A,real),G),topolo7230453075368039082e_nhds(real,im(A2)),F4) ) ).

% tendsto_Im
tff(fact_6496_tendsto__of__int__floor,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_adf(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,ring_1_of_int(C,archim6421214686448440834_floor(B,L))),F4) ) ) ) ).

% tendsto_of_int_floor
tff(fact_6497_tendsto__max,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X5: fun(A,B),Xb: B,Net: filter(A),Y6: fun(A,B),Y: B] :
          ( filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,Xb),Net)
         => ( filterlim(A,B,Y6,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adg(fun(A,B),fun(fun(A,B),fun(A,B)),X5),Y6),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),ord_max(B),Xb),Y)),Net) ) ) ) ).

% tendsto_max
tff(fact_6498_tendsto__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,B,aTP_Lamp_adh(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sinh(B,A2)),F4) ) ) ).

% tendsto_sinh
tff(fact_6499_tendsto__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,B,aTP_Lamp_adi(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,cosh(B,A2)),F4) ) ) ).

% tendsto_cosh
tff(fact_6500_tendsto__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adj(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,real_V8093663219630862766scaleR(B,A2),B2)),F4) ) ) ) ).

% tendsto_scaleR
tff(fact_6501_tendsto__real__root,axiom,
    ! [A: $tType,F2: fun(A,real),Xb: real,F4: filter(A),Nb: nat] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xb),F4)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_adk(fun(A,real),fun(nat,fun(A,real)),F2),Nb),topolo7230453075368039082e_nhds(real,aa(real,real,root(Nb),Xb)),F4) ) ).

% tendsto_real_root
tff(fact_6502_tendsto__of__real__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,real),C2: real,F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_adl(fun(A,real),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),C2)),F4)
        <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4) ) ) ).

% tendsto_of_real_iff
tff(fact_6503_tendsto__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),A2: real,F4: filter(A)] :
          ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,A2),F4)
         => filterlim(A,B,aTP_Lamp_adm(fun(A,real),fun(A,B),G),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),A2)),F4) ) ) ).

% tendsto_of_real
tff(fact_6504_tendsto__arsinh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => filterlim(A,real,aTP_Lamp_adn(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arsinh(real),A2)),F4) ) ).

% tendsto_arsinh
tff(fact_6505_tendsto__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,B,aTP_Lamp_ado(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,exp(B),A2)),F4) ) ) ).

% tendsto_exp
tff(fact_6506_tendsto__real__sqrt,axiom,
    ! [A: $tType,F2: fun(A,real),Xb: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xb),F4)
     => filterlim(A,real,aTP_Lamp_adp(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,Xb)),F4) ) ).

% tendsto_real_sqrt
tff(fact_6507_tendsto__power__strong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,nat),B2: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,nat,G,topolo7230453075368039082e_nhds(nat,B2),F4)
           => filterlim(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_adq(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),B2)),F4) ) ) ) ).

% tendsto_power_strong
tff(fact_6508_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_adr(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Nb)),F4) ) ) ).

% tendsto_power
tff(fact_6509_differentiable__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F4: filter(A)] : differentiable(A,A,aTP_Lamp_yc(A,A),F4) ) ).

% differentiable_ident
tff(fact_6510_differentiable__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: B,F4: filter(A)] : differentiable(A,B,aTP_Lamp_xw(B,fun(A,B),A2),F4) ) ).

% differentiable_const
tff(fact_6511_tendsto__arctan,axiom,
    ! [A: $tType,F2: fun(A,real),Xb: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,Xb),F4)
     => filterlim(A,real,aTP_Lamp_ads(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arctan,Xb)),F4) ) ).

% tendsto_arctan
tff(fact_6512_tendsto__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,real,aTP_Lamp_adb(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,real_V7770717601297561774m_norm(B,A2)),F4) ) ) ).

% tendsto_norm
tff(fact_6513_tendsto__Complex,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => filterlim(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_adt(fun(A,real),fun(fun(A,real),fun(A,complex)),F2),G),topolo7230453075368039082e_nhds(complex,complex2(A2,B2)),F4) ) ) ).

% tendsto_Complex
tff(fact_6514_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,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,B,aTP_Lamp_aaw(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sin(B,A2)),F4) ) ) ).

% tendsto_sin
tff(fact_6515_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,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,B,aTP_Lamp_aax(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,cos(B,A2)),F4) ) ) ).

% tendsto_cos
tff(fact_6516_tendsto__of__int__ceiling,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_adu(fun(A,B),fun(A,C),F2),topolo7230453075368039082e_nhds(C,ring_1_of_int(C,archimedean_ceiling(B,L))),F4) ) ) ) ).

% tendsto_of_int_ceiling
tff(fact_6517_tendsto__Re,axiom,
    ! [A: $tType,G: fun(A,complex),A2: complex,F4: filter(A)] :
      ( filterlim(A,complex,G,topolo7230453075368039082e_nhds(complex,A2),F4)
     => filterlim(A,real,aTP_Lamp_ot(fun(A,complex),fun(A,real),G),topolo7230453075368039082e_nhds(real,re(A2)),F4) ) ).

% tendsto_Re
tff(fact_6518_tendsto__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),F4) ) ) ) ).

% tendsto_mult
tff(fact_6519_tendsto__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_adw(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),L)),F4) ) ) ).

% tendsto_mult_left
tff(fact_6520_tendsto__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_adx(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),L),C2)),F4) ) ) ).

% tendsto_mult_right
tff(fact_6521_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ady(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,one_one(B)),F4) ) ) ) ).

% tendsto_mult_one
tff(fact_6522_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_acv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),F4) ) ) ) ).

% tendsto_add
tff(fact_6523_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,F2: fun(A,B),D2: B,F4: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adz(B,fun(fun(A,B),fun(A,B)),C2),F2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),D2)),F4)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,D2),F4) ) ) ).

% tendsto_add_const_iff
tff(fact_6524_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F4)
         => ( differentiable(A,B,G,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% differentiable_add
tff(fact_6525_tendsto__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,B),B2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aea(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),F4) ) ) ) ).

% tendsto_diff
tff(fact_6526_differentiable__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,F4)
         => ( differentiable(A,B,G,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yg(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),F4) ) ) ) ).

% differentiable_diff
tff(fact_6527_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,C),B2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B2),F4)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aeb(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)),F4) ) ) ) ).

% tendsto_Pair
tff(fact_6528_tendsto__fst,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,product_prod(B,C)),A2: product_prod(B,C),F4: filter(A)] :
          ( filterlim(A,product_prod(B,C),F2,topolo7230453075368039082e_nhds(product_prod(B,C),A2),F4)
         => filterlim(A,B,aTP_Lamp_aec(fun(A,product_prod(B,C)),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(product_prod(B,C),B,product_fst(B,C),A2)),F4) ) ) ).

% tendsto_fst
tff(fact_6529_tendsto__snd,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,product_prod(B,C)),A2: product_prod(B,C),F4: filter(A)] :
          ( filterlim(A,product_prod(B,C),F2,topolo7230453075368039082e_nhds(product_prod(B,C),A2),F4)
         => filterlim(A,C,aTP_Lamp_aed(fun(A,product_prod(B,C)),fun(A,C),F2),topolo7230453075368039082e_nhds(C,aa(product_prod(B,C),C,product_snd(B,C),A2)),F4) ) ) ).

% tendsto_snd
tff(fact_6530_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_aee(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_aef(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ).

% tendsto_null_sum
tff(fact_6531_tendsto__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A2: fun(A,C),F4: filter(B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I3),topolo7230453075368039082e_nhds(C,aa(A,C,A2,I3)),F4) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aeh(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)),F4) ) ) ).

% tendsto_sum
tff(fact_6532_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),one_one(real))
         => filterlim(A,real,aTP_Lamp_aei(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_6533_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_aej(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I3),topolo7230453075368039082e_nhds(C,one_one(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_aek(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,one_one(C)),F4) ) ) ).

% tendsto_one_prod'
tff(fact_6534_tendsto__rabs,axiom,
    ! [A: $tType,F2: fun(A,real),L: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,L),F4)
     => filterlim(A,real,aTP_Lamp_ael(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,abs_abs(real),L)),F4) ) ).

% tendsto_rabs
tff(fact_6535_tendsto__prod_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),F2: fun(A,fun(B,C)),A2: fun(A,C),F4: filter(B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => filterlim(B,C,aa(A,fun(B,C),F2,I3),topolo7230453075368039082e_nhds(C,aa(A,C,A2,I3)),F4) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aen(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),A2),I5)),F4) ) ) ).

% tendsto_prod'
tff(fact_6536_tendsto__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),F2: fun(A,fun(B,C)),L5: fun(A,C),F4: filter(B)] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),S)
             => filterlim(B,C,aa(A,fun(B,C),F2,I3),topolo7230453075368039082e_nhds(C,aa(A,C,L5,I3)),F4) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aep(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2),topolo7230453075368039082e_nhds(C,aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),L5),S)),F4) ) ) ).

% tendsto_prod
tff(fact_6537_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_aeq(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,sgn_sgn(B),L)),F4) ) ) ) ).

% tendsto_sgn
tff(fact_6538_tendsto__rabs__zero__cancel,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_ael(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ).

% tendsto_rabs_zero_cancel
tff(fact_6539_tendsto__rabs__zero__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_ael(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
    <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ).

% tendsto_rabs_zero_iff
tff(fact_6540_tendsto__rabs__zero,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => filterlim(A,real,aTP_Lamp_ael(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ).

% tendsto_rabs_zero
tff(fact_6541_filterlim__at__within__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),G7: filter(B),Xb: A,A3: set(A),P: fun(A,$o),G: fun(A,B)] :
          ( filterlim(A,B,F2,G7,topolo174197925503356063within(A,Xb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),P))))
         => ( filterlim(A,B,G,G7,topolo174197925503356063within(A,Xb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aer(fun(A,$o),fun(A,$o),P)))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_aes(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G7,topolo174197925503356063within(A,Xb,A3)) ) ) ) ).

% filterlim_at_within_If
tff(fact_6542_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aet(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_field_derivative_iff
tff(fact_6543_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,S))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aet(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_field_derivativeD
tff(fact_6544_differentiable__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,real),Xb: A,S3: set(A),G: fun(A,B)] :
          ( differentiable(A,real,F2,topolo174197925503356063within(A,Xb,S3))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xb,S3))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xu(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% differentiable_scaleR
tff(fact_6545_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L: B,A2: A,G: fun(A,C),Mb: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X2: A] :
                ( ( X2 != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(A,C,G,X2)),Mb))),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X2)),L))) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,Mb),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_6546_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,real),A2: A,G: fun(A,real)] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X2: A] :
                ( ( X2 != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,G,X2)) )
           => ( ! [X2: A] :
                  ( ( X2 != A2 )
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,G,X2)),aa(A,real,F2,X2)) )
             => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_6547_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_aeu(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_6548_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_aeu(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_6549_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_aeu(fun(A,B),fun(A,fun(A,B)),F2),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_6550_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K: A,A2: B] :
          ( ( K != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_aev(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_6551_LIM__const__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & topological_t2_space(A) )
     => ! [K: A,L5: A,A2: B] :
          ( ( K != L5 )
         => ~ filterlim(B,A,aTP_Lamp_aew(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,L5),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_const_not_eq
tff(fact_6552_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),F4: 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),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aex(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F4) ) ) ) ).

% tendsto_compose
tff(fact_6553_LIM__const__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [K: B,L5: B,A2: A] :
          ( filterlim(A,B,aTP_Lamp_aey(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( K = L5 ) ) ) ).

% LIM_const_eq
tff(fact_6554_filterlim__at__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),G7: filter(B),Xb: A,P: fun(A,$o),G: fun(A,B)] :
          ( filterlim(A,B,F2,G7,topolo174197925503356063within(A,Xb,aa(fun(A,$o),set(A),collect(A),P)))
         => ( filterlim(A,B,G,G7,topolo174197925503356063within(A,Xb,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aer(fun(A,$o),fun(A,$o),P))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_aes(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G7,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% filterlim_at_If
tff(fact_6555_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,K: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_aez(fun(A,B),fun(A,fun(A,B)),F2),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_6556_tendsto__const__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F4: filter(A),A2: B,B2: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,aTP_Lamp_afa(B,fun(A,B),A2),topolo7230453075368039082e_nhds(B,B2),F4)
          <=> ( A2 = B2 ) ) ) ) ).

% tendsto_const_iff
tff(fact_6557_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_afb(B,fun(fun(B,C),fun(B,C)),A2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_6558_filterlim__INF_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,Xb: A,A3: set(A),F2: fun(B,C),F4: filter(C),G7: fun(A,filter(B))] :
      ( aa(set(A),$o,member(A,Xb),A3)
     => ( filterlim(B,C,F2,F4,aa(A,filter(B),G7,Xb))
       => filterlim(B,C,F2,F4,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),G7),A3))) ) ) ).

% filterlim_INF'
tff(fact_6559_filterlim__INF,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G7: fun(C,filter(B)),B4: set(C),F4: filter(A)] :
      ( filterlim(A,B,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image(C,filter(B),G7),B4)),F4)
    <=> ! [X: C] :
          ( aa(set(C),$o,member(C,X),B4)
         => filterlim(A,B,F2,aa(C,filter(B),G7,X),F4) ) ) ).

% filterlim_INF
tff(fact_6560_differentiable__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F2: fun(A,B),Xb: A,S3: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S3))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xb,S3))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% differentiable_mult
tff(fact_6561_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),Xb: C,S3: set(C)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,aa(C,A,G,Xb),aa(set(C),set(A),image(C,A,G),S3)))
         => ( differentiable(C,A,G,topolo174197925503356063within(C,Xb,S3))
           => differentiable(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo174197925503356063within(C,Xb,S3)) ) ) ) ).

% differentiable_in_compose
tff(fact_6562_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),Xb: C,S3: set(C)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,aa(C,A,G,Xb),top_top(set(A))))
         => ( differentiable(C,A,G,topolo174197925503356063within(C,Xb,S3))
           => differentiable(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo174197925503356063within(C,Xb,S3)) ) ) ) ).

% differentiable_compose
tff(fact_6563_differentiable__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S3: set(A),Nb: nat] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S3))
         => differentiable(A,B,aa(nat,fun(A,B),aTP_Lamp_za(fun(A,B),fun(nat,fun(A,B)),F2),Nb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% differentiable_power
tff(fact_6564_differentiable__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [S3: set(A),F2: fun(A,fun(B,C)),Net: filter(B)] :
          ( finite_finite2(A,S3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S3)
               => differentiable(B,C,aa(A,fun(B,C),F2,X2),Net) )
           => differentiable(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yi(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S3),F2),Net) ) ) ) ).

% differentiable_sum
tff(fact_6565_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [S2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S2)
                & ! [X4: A] :
                    ( ( ( X4 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A2))),S2) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X4)),L5))),R) ) ) ) ) ) ).

% LIM_D
tff(fact_6566_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S8: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S8)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A2))),S8) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X2)),L5))),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_6567_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,X)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_6568_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R2: real,A2: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( ! [X2: A] :
                ( ( X2 != A2 )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A2))),R2)
                 => ( aa(A,B,F2,X2) = aa(A,B,G,X2) ) ) )
           => ( 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_6569_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A2: A,D5: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_afc(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_afd(fun(A,A),fun(A,fun(A,A)),F2),A2),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_6570_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))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X4))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F2,X4)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_6571_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] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X4))),R3) )
               => ( aa(real,real,F2,X4) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_6572_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))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C2),X4))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X4)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_6573_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A2))),D6) )
                     => ( aa(A,B,F2,X2) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_afe(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_6574_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aff(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_6575_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,F2,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aff(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_6576_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_afg(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_6577_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A2))),D6) )
                     => ( aa(A,B,F2,X2) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_afe(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_6578_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S3: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S3))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yr(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% differentiable_divide
tff(fact_6579_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( ! [H3: A] :
                ( ( H3 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,H3))),aa(real,real,aa(real,fun(real,real),times_times(real),K5),real_V7770717601297561774m_norm(A,H3))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_6580_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S3: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S3))
         => ( ( aa(A,B,F2,Xb) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_afh(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% differentiable_inverse
tff(fact_6581_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D5: A,Xb: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D5),topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_aff(fun(A,A),fun(A,fun(A,A)),F2),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_6582_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),A2: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_afi(fun(A,B),fun(A,fun(A,B)),F2),A2),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_6583_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F4: filter(B),A2: A,D2: A] :
          ( filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D2)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_6584_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),D2: A,F4: filter(B),A2: A] :
          ( filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F2),aa(A,fun(A,A),plus_plus(A),D2)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A))))
        <=> filterlim(A,B,F2,F4,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_6585_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
         => ( ! [X2: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X2)),S3)
               => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),A2),X2)),aa(A,A,F2,X2)) )
           => 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_6586_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: real,A2: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
         => ( ! [X2: A] :
                ( ( X2 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X2)),S3)
                 => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),A2),X2)),aa(A,A,F2,X2)) ) )
           => 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_6587_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F2: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( summable(real,F2)
           => ( ! [H3: A,N: nat] :
                  ( ( H3 != zero_zero(A) )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H3),N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N)),real_V7770717601297561774m_norm(A,H3))) ) )
             => filterlim(A,B,aTP_Lamp_afj(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_6588_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_afk(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),aa(num,num,bit0,one2))),top_top(set(real)))) ).

% LIM_cos_div_sin
tff(fact_6589_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)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,A2,zero_zero(nat))),zero_zero(real))
         => ! [N8: nat] : aa(set(real),$o,member(real,suminf(real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),N8)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N8))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_6590_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)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(nat,real,A2,zero_zero(nat)))
         => ! [N8: nat] : aa(set(real),$o,member(real,suminf(real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N8))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),N8)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_6591_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_afm(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_6592_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_afn(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_6593_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_afo(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_6594_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_afp(fun(nat,A),fun(nat,A),F2),F4,at_top(nat))
    <=> filterlim(nat,A,F2,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_6595_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_6596_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ? [U2: fun(nat,A)] :
              ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,U2,N8)),Xb)
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_6597_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [U2: fun(nat,A)] :
              ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(nat,A,U2,N8))
              & filterlim(nat,A,U2,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_6598_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_afq(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_6599_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_afq(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_6600_LIMSEQ__const__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [K: A,L: A] :
          ( filterlim(nat,A,aTP_Lamp_afr(A,fun(nat,A),K),topolo7230453075368039082e_nhds(A,L),at_top(nat))
        <=> ( K = L ) ) ) ).

% LIMSEQ_const_iff
tff(fact_6601_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_afs(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_6602_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_afs(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_6603_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_aft(fun(nat,A),fun(nat,fun(nat,A)),F2),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_6604_differentiable__cnj__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,complex),Xb: A,A3: set(A)] :
          ( differentiable(A,complex,aTP_Lamp_aad(fun(A,complex),fun(A,complex),F2),topolo174197925503356063within(A,Xb,A3))
        <=> differentiable(A,complex,F2,topolo174197925503356063within(A,Xb,A3)) ) ) ).

% differentiable_cnj_iff
tff(fact_6605_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,A2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N6: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_6606_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,A2: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N6: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,X5,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb) ) ) ) ).

% LIMSEQ_le_const
tff(fact_6607_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,N2: nat,C5: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),aa(nat,A,F2,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),L) ) ) ) ).

% Lim_bounded2
tff(fact_6608_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,M6: nat,C5: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),C5) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),C5) ) ) ) ).

% Lim_bounded
tff(fact_6609_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,Y6: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N6: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,Y6,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ) ).

% LIMSEQ_le
tff(fact_6610_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N2: nat,X5: fun(nat,A),Y6: fun(nat,A),Xb: A,Y: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,Y6,N)) )
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
           => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ) ).

% lim_mono
tff(fact_6611_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S3: set(A),A2: A] :
          ( ! [N: nat] : aa(set(A),$o,member(A,aa(nat,A,B2,N)),S3)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,complete_Sup_Sup(A),S3)) ) ) ) ).

% Sup_lim
tff(fact_6612_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S3: set(A),A2: A] :
          ( ! [N: nat] : aa(set(A),$o,member(A,aa(nat,A,B2,N)),S3)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S3)),A2) ) ) ) ).

% Inf_lim
tff(fact_6613_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C2)
     => filterlim(nat,nat,aTP_Lamp_afu(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_6614_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_6615_monoseq__convergent,axiom,
    ! [X5: fun(nat,real),B4: real] :
      ( topological_monoseq(real,X5)
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,X5,I3))),B4)
       => ~ ! [L6: real] : ~ filterlim(nat,real,X5,topolo7230453075368039082e_nhds(real,L6),at_top(nat)) ) ) ).

% monoseq_convergent
tff(fact_6616_LIMSEQ__lessThan__iff__atMost,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(set(nat),A),Xb: A] :
          ( filterlim(nat,A,aTP_Lamp_afv(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
        <=> filterlim(nat,A,aTP_Lamp_afw(fun(set(nat),A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ).

% LIMSEQ_lessThan_iff_atMost
tff(fact_6617_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_afx(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_6618_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: fun(nat,A),Xb: A] :
          ( topological_monoseq(A,A2)
         => ( filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
           => ( ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,N8)),Xb)
                & ! [M2: nat,N8: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N8)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,M2)),aa(nat,A,A2,N8)) ) )
              | ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(nat,A,A2,N8))
                & ! [M2: nat,N8: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N8)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,N8)),aa(nat,A,A2,M2)) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_6619_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_afy(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_6620_LIMSEQ__SEQ__conv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,X5: fun(A,B),L5: B] :
          ( ! [S9: fun(nat,A)] :
              ( ( ! [N4: nat] : aa(nat,A,S9,N4) != A2
                & filterlim(nat,A,S9,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_afz(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S9),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) )
        <=> filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv
tff(fact_6621_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))))
         => ! [S10: fun(nat,A)] :
              ( ( ! [N: nat] : aa(nat,A,S10,N) != A2
                & filterlim(nat,A,S10,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aga(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S10),topolo7230453075368039082e_nhds(B,L),at_top(nat)) ) ) ) ).

% LIMSEQ_SEQ_conv1
tff(fact_6622_LIMSEQ__SEQ__conv2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F2: fun(A,B),L: B] :
          ( ! [S5: fun(nat,A)] :
              ( ( ! [N8: nat] : aa(nat,A,S5,N8) != A2
                & filterlim(nat,A,S5,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_afz(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),S5),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_6623_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_agb(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_6624_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X5: fun(nat,A),Xb: A,L: nat] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_agc(fun(nat,A),fun(nat,fun(nat,A)),X5),L),topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_6625_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_agd(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable'
tff(fact_6626_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_age(fun(nat,A),fun(nat,A),F2)) ) ) ).

% telescope_summable
tff(fact_6627_nested__sequence__unique,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,G,aa(nat,nat,suc,N))),aa(nat,real,G,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,G,N))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_agf(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L3: real] :
                ( ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N8)),L3)
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L3),at_top(nat))
                & ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L3),aa(nat,real,G,N8))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_6628_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N6: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R3),aa(nat,real,X5,N)) )
     => filterlim(nat,real,aTP_Lamp_agg(fun(nat,real),fun(nat,real),X5),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_6629_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_agh(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_6630_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
     => filterlim(nat,real,aTP_Lamp_agi(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_6631_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_agj(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_6632_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_agk(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_6633_sums__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A] :
          ( aa(A,$o,sums(A,F2),S3)
        <=> filterlim(nat,A,aTP_Lamp_agl(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S3),at_top(nat)) ) ) ).

% sums_def
tff(fact_6634_sums__def__le,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A] :
          ( aa(A,$o,sums(A,F2),S3)
        <=> filterlim(nat,A,aTP_Lamp_agm(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S3),at_top(nat)) ) ) ).

% sums_def_le
tff(fact_6635_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),aa(nat,real,F2,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N)),L)
       => ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,N8)),E2)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_6636_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_agn(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_6637_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_ago(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_6638_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_agp(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_6639_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => aa(A,$o,sums(A,aTP_Lamp_agd(fun(nat,A),fun(nat,A),F2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F2,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_6640_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => aa(A,$o,sums(A,aTP_Lamp_age(fun(nat,A),fun(nat,A),F2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_6641_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
       => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_6642_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_agq(real,fun(real,fun(nat,real)),Xb),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_6643_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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_6644_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),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_6645_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => filterlim(nat,real,aTP_Lamp_agr(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_6646_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A] :
          ( aa(A,$o,sums(A,F2),S3)
        <=> filterlim(nat,A,aTP_Lamp_ags(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S3),at_top(nat)) ) ) ).

% sums_def'
tff(fact_6647_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),Xb: real] :
          ( filterlim(nat,real,aTP_Lamp_agt(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,Xb),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_6648_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_agu(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_6649_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_agv(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ
tff(fact_6650_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_agw(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,suminf(A,F2)),at_top(nat)) ) ) ).

% summable_LIMSEQ'
tff(fact_6651_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No: nat] :
                ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,N4)),L5))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_6652_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,N)),L5))),R3) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_6653_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [No3: nat] :
              ! [N8: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N8)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X5,N8)),L5))),R) ) ) ) ) ).

% LIMSEQ_D
tff(fact_6654_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_6655_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim(nat,real,aTP_Lamp_agx(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xb)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_6656_tendsto__at__iff__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: B,Xb: A,S3: set(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),topolo174197925503356063within(A,Xb,S3))
        <=> ! [X7: fun(nat,A)] :
              ( ! [I4: nat] : aa(set(A),$o,member(A,aa(nat,A,X7,I4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))
             => ( filterlim(nat,A,X7,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
               => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),comp(A,B,nat,F2),X7),topolo7230453075368039082e_nhds(B,A2),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_6657_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F4: filter(A),Xb: B] :
          ( filterlim(A,nat,F2,at_top(nat),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,Xb)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_agy(fun(A,nat),fun(B,fun(A,B)),F2),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_6658_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] : filterlim(nat,real,aTP_Lamp_agz(real,fun(nat,real),R),topolo7230453075368039082e_nhds(real,R),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_6659_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N))),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,N))))
         => filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_6660_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_afl(fun(nat,real),fun(nat,real),A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_6661_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))
           => ( ! [N: nat] : aa(nat,A,S3,N) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_aha(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_6662_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_ahb(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_6663_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xb))
         => filterlim(nat,A,aTP_Lamp_ahc(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_6664_summable,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => summable(real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_6665_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_ahd(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K2: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ahe(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_6666_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_ahf(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K2: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ahe(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_6667_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_ahg(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_6668_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_dm(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_6669_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun(nat,real),Nb: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))),suminf(real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_6670_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => filterlim(nat,real,aTP_Lamp_ahg(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_6671_sums__alternating__upper__lower,axiom,
    ! [A2: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L3: real] :
              ( ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N8)))),L3)
              & filterlim(nat,real,aTP_Lamp_ahg(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat))
              & ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L3),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),N8)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_ahh(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_6672_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_ahh(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_6673_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),Nb: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),Nb)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_6674_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => filterlim(nat,real,aTP_Lamp_ahh(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_6675_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ahi(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_6676_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D5: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D5)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_ahj(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D5),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_6677_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,F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,A2),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahk(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),F4) ) ) ) ).

% bounded_linear.tendsto
tff(fact_6678_bounded__linear_Ocontinuous,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(C),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3448309680560233919inuous(C,A,F4,G)
           => topolo3448309680560233919inuous(C,B,F4,aa(fun(C,A),fun(C,B),aTP_Lamp_ahl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.continuous
tff(fact_6679_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_ahm(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_6680_real__bounded__linear,axiom,
    ! [F2: fun(real,real)] :
      ( real_V3181309239436604168linear(real,real,F2)
    <=> ? [C4: real] :
        ! [X: real] : aa(real,real,F2,X) = aa(real,real,aa(real,fun(real,real),times_times(real),X),C4) ) ).

% real_bounded_linear
tff(fact_6681_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_xx(A,B)) ) ).

% bounded_linear_zero
tff(fact_6682_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_xy(fun(A,B),fun(A,B),F2)) ) ) ).

% bounded_linear_minus
tff(fact_6683_bounded__linear_OCauchy,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( topolo3814608138187158403Cauchy(A,X5)
           => topolo3814608138187158403Cauchy(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ahn(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ).

% bounded_linear.Cauchy
tff(fact_6684_bounded__linear_Osums,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A),A2: A] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( aa(A,$o,sums(A,X5),A2)
           => aa(B,$o,sums(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ahn(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)),aa(A,B,F2,A2)) ) ) ) ).

% bounded_linear.sums
tff(fact_6685_bounded__linear__scaleR__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,real),Xb: B] :
          ( real_V3181309239436604168linear(A,real,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_ya(fun(A,real),fun(B,fun(A,B)),G),Xb)) ) ) ).

% bounded_linear_scaleR_const
tff(fact_6686_bounded__linear__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : real_V3181309239436604168linear(real,A,aTP_Lamp_aho(A,fun(real,A),Xb)) ) ).

% bounded_linear_scaleR_left
tff(fact_6687_bounded__linear__const__scaleR,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),R: real] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(real,fun(A,B),aTP_Lamp_xz(fun(A,B),fun(real,fun(A,B)),G),R)) ) ) ).

% bounded_linear_const_scaleR
tff(fact_6688_bounded__linear__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R: real] : real_V3181309239436604168linear(A,A,real_V8093663219630862766scaleR(A,R)) ) ).

% bounded_linear_scaleR_right
tff(fact_6689_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_yl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear_compose
tff(fact_6690_bounded__linear__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => real_V3181309239436604168linear(A,A,aTP_Lamp_yc(A,A)) ) ).

% bounded_linear_ident
tff(fact_6691_bounded__linear_Osummable,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X5)
           => summable(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ahn(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ).

% bounded_linear.summable
tff(fact_6692_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_6693_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_ahp(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_6694_bounded__linear__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Xb: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_yd(fun(A,B),fun(B,fun(A,B)),G),Xb)) ) ) ).

% bounded_linear_const_mult
tff(fact_6695_bounded__linear__mult__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Y: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_ye(fun(A,B),fun(B,fun(A,B)),G),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_6696_bounded__linear__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A] : real_V3181309239436604168linear(A,A,aa(A,fun(A,A),times_times(A),Xb)) ) ).

% bounded_linear_mult_right
tff(fact_6697_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_yf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_add
tff(fact_6698_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_yg(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_6699_bounded__linear__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => real_V3181309239436604168linear(B,C,aa(A,fun(B,C),F2,I3)) )
         => real_V3181309239436604168linear(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_yi(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% bounded_linear_sum
tff(fact_6700_bounded__linear_Osuminf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( summable(A,X5)
           => ( aa(A,B,F2,suminf(A,X5)) = suminf(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ahn(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ) ).

% bounded_linear.suminf
tff(fact_6701_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),G5: fun(C,A),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( has_derivative(C,A,G,G5,F4)
           => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),aa(fun(C,A),fun(C,B),aTP_Lamp_yl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G5),F4) ) ) ) ).

% bounded_linear.has_derivative
tff(fact_6702_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K9: real] :
            ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),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_6703_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),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahk(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_6704_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_ahl(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.isCont
tff(fact_6705_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K9: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),K9)
              & ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),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_6706_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K9: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K9)
              & ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),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_6707_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K5: real] :
          ( ! [X2: A,Y4: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4))
         => ( ! [R3: real,X2: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R3),X2)) = aa(B,B,real_V8093663219630862766scaleR(B,R3),aa(A,B,F2,X2))
           => ( ! [X2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K5))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_6708_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_ahq(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),F6),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_6709_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F6: fun(A,B),Xb: A,F2: fun(A,B),S3: set(A)] :
          ( real_V3181309239436604168linear(A,B,F6)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_ahr(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F6),Xb),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S3))
           => has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivativeI
tff(fact_6710_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ahs(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_at_within
tff(fact_6711_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & ? [E4: fun(A,B)] :
                ( ! [H5: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5)) = 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,Xb)),aa(A,B,F6,H5))),aa(A,B,E4,H5))
                & filterlim(A,real,aTP_Lamp_aht(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_6712_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ahi(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F6),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% has_derivative_within
tff(fact_6713_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F2,F6,F4)
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_ahu(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F6),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_6714_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xb: A,S: set(A),F2: fun(A,B),F6: fun(A,B)] :
          ( aa(set(A),$o,member(A,Xb),S)
         => ( aa(set(A),$o,topolo1002775350975398744n_open(A),S)
           => ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S))
            <=> ( real_V3181309239436604168linear(A,B,F6)
                & ? [E4: fun(A,B)] :
                    ( ! [H5: A] :
                        ( aa(set(A),$o,member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5)),S)
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5)) = 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,Xb)),aa(A,B,F6,H5))),aa(A,B,E4,H5)) ) )
                    & filterlim(A,real,aTP_Lamp_aht(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_6715_lim__const,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_afr(A,fun(nat,A),A2)) = A2 ) ).

% lim_const
tff(fact_6716_open__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A3: set(A),B4: fun(A,set(B))] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(set(B),$o,topolo1002775350975398744n_open(B),aa(A,set(B),B4,X2)) )
         => aa(set(B),$o,topolo1002775350975398744n_open(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% open_UN
tff(fact_6717_open__INT,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A3: set(A),B4: fun(A,set(B))] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => aa(set(B),$o,topolo1002775350975398744n_open(B),aa(A,set(B),B4,X2)) )
           => aa(set(B),$o,topolo1002775350975398744n_open(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ) ).

% open_INT
tff(fact_6718_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Xb) )
           => ~ aa(set(A),$o,member(A,aa(set(A),A,complete_Sup_Sup(A),A3)),A3) ) ) ) ).

% Sup_notin_open
tff(fact_6719_open__Collect__conj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),P))
         => ( aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),Q))
           => aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_conj
tff(fact_6720_open__Collect__disj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),P))
         => ( aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),Q))
           => aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahw(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_disj
tff(fact_6721_open__Collect__const,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: $o] : aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ahx($o,fun(A,$o),(P)))) ) ).

% open_Collect_const
tff(fact_6722_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),Xb: A] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X2) )
           => ~ aa(set(A),$o,member(A,aa(set(A),A,complete_Inf_Inf(A),A3)),A3) ) ) ) ).

% Inf_notin_open
tff(fact_6723_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),Xb: A,Y: A] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),S)
         => ( aa(set(A),$o,member(A,Xb),S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
             => ? [B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B3)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Xb,B3)),S) ) ) ) ) ) ).

% open_right
tff(fact_6724_Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,S3: set(A)] :
          ( ( topolo174197925503356063within(A,Xb,S3) != bot_bot(filter(A)) )
         => ( topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,Xb,S3),aTP_Lamp_aac(A,A)) = Xb ) ) ) ).

% Lim_ident_at
tff(fact_6725_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S9: set(A)] :
              ( aa(set(A),$o,topolo1002775350975398744n_open(A),S9)
             => ( aa(set(A),$o,member(A,F0),S9)
               => ? [N5: nat] :
                  ! [N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N4)
                   => aa(set(A),$o,member(A,aa(nat,A,F2,N4)),S9) ) ) ) ) ) ).

% lim_explicit
tff(fact_6726_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_abc(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_6727_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_abf(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_6728_continuous__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
        <=> filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A)))),F4) ) ) ).

% continuous_def
tff(fact_6729_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_abg(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_6730_t2__space__class_OLim__def,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [A3: filter(B),F2: fun(B,A)] : topolo3827282254853284352ce_Lim(B,A,A3,F2) = the(A,aa(fun(B,A),fun(A,$o),aTP_Lamp_ahy(filter(B),fun(fun(B,A),fun(A,$o)),A3),F2)) ) ).

% t2_space_class.Lim_def
tff(fact_6731_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A,S: set(A),T2: set(A),U3: set(A)] :
          ( aa(set(A),$o,member(A,Xb),S)
         => ( aa(set(A),$o,topolo1002775350975398744n_open(A),S)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T2),S)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U3),S)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,Xb,T2) = topolo174197925503356063within(A,Xb,U3) ) ) ) ) ) ).

% at_within_nhd
tff(fact_6732_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,real,F4,G)
           => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_abz(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_powr
tff(fact_6733_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aca(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_ln
tff(fact_6734_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_agv(fun(nat,A),fun(nat,A),F2)) ) ).

% suminf_eq_lim
tff(fact_6735_lim__def,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X5: fun(nat,A)] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),X5) = the(A,aTP_Lamp_ahz(fun(nat,A),fun(A,$o),X5)) ) ).

% lim_def
tff(fact_6736_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( cos(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aia(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_abi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_6737_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F4: filter(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F4,F2)
         => ( ( sin(A,aa(A,A,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aia(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F4,aTP_Lamp_abj(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_6738_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( cosh(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aTP_Lamp_abk(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_tanh
tff(fact_6739_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aib(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_6740_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,S: set(B),F2: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( aa(set(B),$o,member(B,A2),S)
           => ( aa(set(B),$o,topolo1002775350975398744n_open(B),S)
             => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,S))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_afb(B,fun(fun(B,C),fun(B,C)),A2),F2),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_6741_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( topolo3448309680560233919inuous(A,real,F4,G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))))
                 => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_acb(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_6742_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( aa(set(real),$o,member(real,aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A)))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aic(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_artanh
tff(fact_6743_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E: real,F6: fun(A,B),S3: set(A),Xb: A,F2: fun(A,B),H6: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
         => ( real_V3181309239436604168linear(A,B,F6)
           => ( ! [Y4: A] :
                  ( aa(set(A),$o,member(A,Y4),S3)
                 => ( ( Y4 != Xb )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y4,Xb)),E)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F2,Y4)),aa(A,B,F2,Xb))),aa(A,B,F6,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y4),Xb))))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y4),Xb)))),aa(A,real,H6,Y4)) ) ) )
             => ( filterlim(A,real,H6,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S3))
               => has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_6744_tendsto__exp__limit__at__right,axiom,
    ! [Xb: real] : filterlim(real,real,aTP_Lamp_aid(real,fun(real,real),Xb),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xb)),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_6745_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( aa(set(A),$o,member(A,I2),aa(A,set(A),set_ord_greaterThan(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),I2) ) ) ).

% greaterThan_iff
tff(fact_6746_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_6747_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_6748_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),Xb)),aa(A,set(A),set_ord_greaterThan(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% greaterThan_subset_iff
tff(fact_6749_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Y))
        <=> ( Xb != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_6750_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xb,Y)),zero_zero(real))
        <=> ( Xb = Y ) ) ) ).

% dist_le_zero_iff
tff(fact_6751_dist__diff_I1_J,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : real_V557655796197034286t_dist(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = real_V7770717601297561774m_norm(A,B2) ) ).

% dist_diff(1)
tff(fact_6752_dist__diff_I2_J,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : real_V557655796197034286t_dist(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),A2) = real_V7770717601297561774m_norm(A,B2) ) ).

% dist_diff(2)
tff(fact_6753_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),top_top(A))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),Xb)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_6754_dist__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: real,A2: A,Y: real] : real_V557655796197034286t_dist(A,aa(A,A,real_V8093663219630862766scaleR(A,Xb),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,aa(real,fun(real,real),minus_minus(real),Xb),Y))),real_V7770717601297561774m_norm(A,A2)) ) ).

% dist_scaleR
tff(fact_6755_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,D2: real] : aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_aie(A,fun(real,fun(A,$o)),Xb),D2))) ) ).

% open_ball
tff(fact_6756_continuous__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F4: filter(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( topolo3448309680560233919inuous(A,B,F4,G)
           => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,B),fun(A,real),aTP_Lamp_aif(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G)) ) ) ) ).

% continuous_dist
tff(fact_6757_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Y)) ) ).

% zero_le_dist
tff(fact_6758_dist__triangle,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Z: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xb,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xb,Y)),real_V557655796197034286t_dist(A,Y,Z))) ) ).

% dist_triangle
tff(fact_6759_dist__triangle2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A,Z: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xb,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xb,Z)),real_V557655796197034286t_dist(A,Y,Z))) ) ).

% dist_triangle2
tff(fact_6760_dist__triangle3,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A,A2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xb,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,A2,Xb)),real_V557655796197034286t_dist(A,A2,Y))) ) ).

% dist_triangle3
tff(fact_6761_dist__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Z: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xb,Z)),real_V557655796197034286t_dist(A,Y,Z))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xb,Y)),E) ) ) ).

% dist_triangle_le
tff(fact_6762_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Y: A,E1: real,X23: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,Y)),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X23,Y)),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X23)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_6763_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Z: A,Y: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xb,Z)),real_V557655796197034286t_dist(A,Y,Z))),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Y)),E) ) ) ).

% dist_triangle_lt
tff(fact_6764_dist__norm,axiom,
    ! [A: $tType] :
      ( real_V6936659425649961206t_norm(A)
     => ! [Xb: A,Y: A] : real_V557655796197034286t_dist(A,Xb,Y) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) ) ).

% dist_norm
tff(fact_6765_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,Xb: A,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,Xb)),E)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Y)),E) ) ) ).

% dist_commute_lessI
tff(fact_6766_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Y)) ) ) ).

% dist_pos_lt
tff(fact_6767_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xb,Y)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_6768_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_6769_dist__real__def,axiom,
    ! [Xb: real,Y: real] : real_V557655796197034286t_dist(real,Xb,Y) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y)) ).

% dist_real_def
tff(fact_6770_dist__complex__def,axiom,
    ! [Xb: complex,Y: complex] : real_V557655796197034286t_dist(complex,Xb,Y) = real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),Xb),Y)) ).

% dist_complex_def
tff(fact_6771_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),S)
        <=> ! [X: A] :
              ( aa(set(A),$o,member(A,X),S)
             => ? [E4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
                  & ! [Y3: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y3,X)),E4)
                     => aa(set(A),$o,member(A,Y3),S) ) ) ) ) ) ).

% open_dist
tff(fact_6772_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [A2: A,B2: A,C2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(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_6773_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( topolo174197925503356063within(A,A2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)) ) ) ) ).

% at_within_Icc_at_right
tff(fact_6774_filterlim__at__left__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A),A2: real] :
      ( filterlim(real,A,F2,F4,topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
    <=> filterlim(real,A,aTP_Lamp_aig(fun(real,A),fun(real,A),F2),F4,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,uminus_uminus(real),A2)))) ) ).

% filterlim_at_left_to_right
tff(fact_6775_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F6: A,A2: A,S: set(A),D2: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F6,topolo174197925503356063within(A,A2,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( aa(set(A),$o,member(A,A2),S)
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),D2)
                     => ( aa(A,A,F2,X2) = aa(A,A,G,X2) ) ) )
               => has_field_derivative(A,G,F6,topolo174197925503356063within(A,A2,S)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_6776_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),D2: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( aa(set(A),$o,member(A,Xb),S3)
             => ( ! [X8: A] :
                    ( aa(set(A),$o,member(A,X8),S3)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xb)),D2)
                     => ( aa(A,B,F2,X8) = aa(A,B,G,X8) ) ) )
               => has_derivative(A,B,G,F6,topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_6777_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M9: nat] :
                ! [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M),aa(nat,A,X5,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% metric_CauchyI
tff(fact_6778_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),E: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => ? [M7: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),M2)
               => ! [N8: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N8)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M2),aa(nat,A,X5,N8))),E) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_6779_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S3: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S3)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [N5: nat] :
                ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S3,N4),aa(nat,A,S3,N5))),E4) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_6780_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M8: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M5)
                 => ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N4)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M5),aa(nat,A,X5,N4))),E4) ) ) ) ) ) ).

% Cauchy_def
tff(fact_6781_dist__of__int,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Mb: int,Nb: int] : real_V557655796197034286t_dist(A,ring_1_of_int(A,Mb),ring_1_of_int(A,Nb)) = ring_1_of_int(real,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),Nb))) ) ).

% dist_of_int
tff(fact_6782_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [A4: A,B3: A] :
              ( aa(set(A),$o,member(A,Xb),aa(A,set(A),set_ord_lessThan(A),A4))
              & aa(set(A),$o,member(A,Y),aa(A,set(A),set_ord_greaterThan(A),B3))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),A4)),aa(A,set(A),set_ord_greaterThan(A),B3)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_6783_tendsto__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),G: fun(A,B),Mb: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Mb),F4)
           => filterlim(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_aih(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,real_V557655796197034286t_dist(B,L,Mb)),F4) ) ) ) ).

% tendsto_dist
tff(fact_6784_filterlim__at__right__to__0,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A),A2: real] :
      ( filterlim(real,A,F2,F4,topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
    <=> filterlim(real,A,aa(real,fun(real,A),aTP_Lamp_aii(fun(real,A),fun(real,fun(real,A)),F2),A2),F4,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% filterlim_at_right_to_0
tff(fact_6785_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Y: A,E: real,X23: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X23,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X23)),E) ) ) ) ).

% dist_triangle_half_l
tff(fact_6786_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X1: A,E: real,X23: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X1)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X23)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X23)),E) ) ) ) ).

% dist_triangle_half_r
tff(fact_6787_metric__LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L: B,A2: A,G: fun(A,C),Mb: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X2: A] :
                ( ( X2 != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,G,X2),Mb)),real_V557655796197034286t_dist(B,aa(A,B,F2,X2),L)) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,Mb),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% metric_LIM_imp_LIM
tff(fact_6788_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,Xb: A,S: set(A),D2: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xb,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( ! [X8: A] :
                  ( aa(set(A),$o,member(A,X8),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X8,Xb))
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xb)),D2)
                     => ( aa(A,B,F2,X8) = aa(A,B,G,X8) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% Lim_transform_within
tff(fact_6789_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,X23: A,E: real,X32: A,X42: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X23)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X23,X32)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X42)),E) ) ) ) ) ).

% dist_triangle_third
tff(fact_6790_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G7: filter(B),Xb: A,S: set(A),F4: filter(B),D2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G7,topolo174197925503356063within(A,Xb,S))
         => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G7),F4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
             => ( ! [X8: A] :
                    ( aa(set(A),$o,member(A,X8),S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X8,Xb))
                     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xb)),D2)
                       => ( aa(A,B,F2,X8) = aa(A,B,G,X8) ) ) ) )
               => filterlim(A,B,F2,F4,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_6791_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M8: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M5)
                 => ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N4)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F2,M5),aa(nat,A,F2,N4))),E4) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_6792_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M9: nat] :
                ! [M: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M),aa(nat,A,X5,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI'
tff(fact_6793_dist__of__nat,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Mb: nat,Nb: nat] : real_V557655796197034286t_dist(A,aa(nat,A,semiring_1_of_nat(A),Mb),aa(nat,A,semiring_1_of_nat(A),Nb)) = ring_1_of_int(real,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Mb)),aa(nat,int,semiring_1_of_nat(int),Nb)))) ) ).

% dist_of_nat
tff(fact_6794_tendsto__dist__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> filterlim(A,real,aa(B,fun(A,real),aTP_Lamp_aij(fun(A,B),fun(B,fun(A,real)),F2),L),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_dist_iff
tff(fact_6795_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),P2: B,F12: filter(A),C2: B,L: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,P2,aa(B,set(B),set_ord_greaterThan(B),P2)),F12)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C2)
           => ( ( L = aa(B,B,aa(B,fun(B,B),times_times(B),C2),P2) )
             => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_aik(fun(A,B),fun(B,fun(A,B)),F2),C2),topolo174197925503356063within(B,L,aa(B,set(B),set_ord_greaterThan(B),L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_6796_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A2: A,R2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ( ! [X2: A] :
                  ( ( X2 != A2 )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),R2)
                   => ( aa(A,B,F2,X2) = aa(A,B,G,X2) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_6797_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F2: fun(A,B),L5: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S8: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S8)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),S8) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X2),L5)),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_6798_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A,R: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [S2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S2)
                & ! [X4: A] :
                    ( ( ( X4 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),S2) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X4),L5)),R) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_6799_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X),L5)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_6800_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A,R: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ? [No3: nat] :
              ! [N8: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N8)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N8),L5)),R) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_6801_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N),L5)),R3) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_6802_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No: nat] :
                ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N4),L5)),R5) ) ) ) ) ).

% lim_sequentially
tff(fact_6803_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [J3: nat] :
            ? [M8: nat] :
            ! [M5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M5)
             => ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N4)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,M5),aa(nat,A,X5,N4))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_6804_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),D6) )
                     => ( aa(A,B,F2,X2) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ail(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_6805_tendsto__arcosh__at__left__1,axiom,
    filterlim(real,real,arcosh(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,one_one(real),aa(real,set(real),set_ord_greaterThan(real),one_one(real)))) ).

% tendsto_arcosh_at_left_1
tff(fact_6806_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A2),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),D6) )
                     => ( aa(A,B,F2,X2) != aa(A,B,F2,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ail(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_6807_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,aa(A,set(A),set_ord_lessThan(A),A2)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),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_aim(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_6808_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),No)
                  & ! [N4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N4)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X5,N4),L5)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_6809_totally__bounded__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: set(A)] :
          ( topolo6688025880775521714ounded(A,S)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [K3: set(A)] :
                  ( finite_finite2(A,K3)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image(A,set(A),aTP_Lamp_aio(real,fun(A,set(A)),E4)),K3))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_6810_at__within__order,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,S3: set(A)] :
          ( ( top_top(set(A)) != aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) )
         => ( topolo174197925503356063within(A,Xb,S3) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aip(A,fun(set(A),fun(A,filter(A))),Xb),S3)),aa(A,set(A),set_ord_greaterThan(A),Xb)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aiq(A,fun(set(A),fun(A,filter(A))),Xb),S3)),aa(A,set(A),set_ord_lessThan(A),Xb)))) ) ) ) ).

% at_within_order
tff(fact_6811_SUP__principal,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B)] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),aTP_Lamp_air(fun(B,set(A)),fun(B,filter(A)),A3)),I5)) = aa(set(A),filter(A),principal(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))) ).

% SUP_principal
tff(fact_6812_filterlim__If,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G7: filter(B),F4: filter(A),P: fun(A,$o),G: fun(A,B)] :
      ( filterlim(A,B,F2,G7,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),P))))
     => ( filterlim(A,B,G,G7,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fy(fun(A,$o),fun(A,$o),P)))))
       => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_ais(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G7,F4) ) ) ).

% filterlim_If
tff(fact_6813_nhds__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] : topolo7230453075368039082e_nhds(A,A2) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(set(A)),set(filter(A)),image(set(A),filter(A),principal(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ait(A,fun(set(A),$o),A2)))) ) ).

% nhds_def
tff(fact_6814_filterlim__base,axiom,
    ! [B: $tType,A: $tType,E3: $tType,D: $tType,C: $tType,J4: set(A),I2: fun(A,C),I5: set(C),F4: fun(C,set(D)),F2: fun(D,E3),G7: fun(A,set(E3))] :
      ( ! [M: A,X2: B] :
          ( aa(set(A),$o,member(A,M),J4)
         => aa(set(C),$o,member(C,aa(A,C,I2,M)),I5) )
     => ( ! [M: A,X2: D] :
            ( aa(set(A),$o,member(A,M),J4)
           => ( aa(set(D),$o,member(D,X2),aa(C,set(D),F4,aa(A,C,I2,M)))
             => aa(set(E3),$o,member(E3,aa(D,E3,F2,X2)),aa(A,set(E3),G7,M)) ) )
       => filterlim(D,E3,F2,aa(set(filter(E3)),filter(E3),complete_Inf_Inf(filter(E3)),aa(set(A),set(filter(E3)),image(A,filter(E3),aTP_Lamp_aiu(fun(A,set(E3)),fun(A,filter(E3)),G7)),J4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(C),set(filter(D)),image(C,filter(D),aTP_Lamp_aiv(fun(C,set(D)),fun(C,filter(D)),F4)),I5))) ) ) ).

% filterlim_base
tff(fact_6815_greaterThan__0,axiom,
    aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_6816_greaterThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_6817_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I5: set(A),F4: fun(A,set(B)),F2: fun(B,C),G7: fun(D,set(C)),J4: set(D)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( aa(set(A),$o,member(A,I3),I5)
           => ! [J2: A] :
                ( aa(set(A),$o,member(A,J2),I5)
               => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,I3)),aa(A,set(B),F4,J2))
                  | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F4,J2)),aa(A,set(B),F4,I3)) ) ) )
       => ( filterlim(B,C,F2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image(D,filter(C),aTP_Lamp_aiw(fun(D,set(C)),fun(D,filter(C)),G7)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),aTP_Lamp_aix(fun(A,set(B)),fun(A,filter(B)),F4)),I5)))
        <=> ! [X: D] :
              ( aa(set(D),$o,member(D,X),J4)
             => ? [Xa2: A] :
                  ( aa(set(A),$o,member(A,Xa2),I5)
                  & ! [Xb4: B] :
                      ( aa(set(B),$o,member(B,Xb4),aa(A,set(B),F4,Xa2))
                     => aa(set(C),$o,member(C,aa(B,C,F2,Xb4)),aa(D,set(C),G7,X)) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_6818_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X5: set(A),F2: fun(A,set(B))] :
      ( finite_finite2(A,X5)
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),aTP_Lamp_aix(fun(A,set(B)),fun(A,filter(B)),F2)),X5)) = aa(set(B),filter(B),principal(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),F2),X5))) ) ) ).

% INF_principal_finite
tff(fact_6819_INT__greaterThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_greaterThan(nat)),top_top(set(nat)))) = bot_bot(set(nat)) ).

% INT_greaterThan_UNIV
tff(fact_6820_at__within__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S3: set(A)] : topolo174197925503356063within(A,A2,S3) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A2)),aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ).

% at_within_def
tff(fact_6821_nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A] : topolo7230453075368039082e_nhds(A,Xb) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(real),set(filter(A)),image(real,filter(A),aTP_Lamp_aiz(A,fun(real,filter(A)),Xb)),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% nhds_metric
tff(fact_6822_at__left__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_lessThan(A),Xb)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_aja(A,fun(A,filter(A)),Xb)),aa(A,set(A),set_ord_lessThan(A),Xb))) ) ) ) ).

% at_left_eq
tff(fact_6823_at__right__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_greaterThan(A),Xb)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_ajb(A,fun(A,filter(A)),Xb)),aa(A,set(A),set_ord_greaterThan(A),Xb))) ) ) ) ).

% at_right_eq
tff(fact_6824_nhds__order,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Xb: A] : topolo7230453075368039082e_nhds(A,Xb) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_ajc(A,filter(A))),aa(A,set(A),set_ord_greaterThan(A),Xb)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_ajd(A,filter(A))),aa(A,set(A),set_ord_lessThan(A),Xb)))) ) ).

% nhds_order
tff(fact_6825_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A,S3: set(A)] : topolo174197925503356063within(A,Xb,S3) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(set(A)),set(filter(A)),image(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_aje(A,fun(set(A),fun(set(A),filter(A))),Xb),S3)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ait(A,fun(set(A),$o),Xb)))) ) ).

% at_within_eq
tff(fact_6826_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),aa(num,num,bit0,one2)))),aa(real,set(real),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),aa(num,num,bit0,one2))))))) ).

% filterlim_tan_at_right
tff(fact_6827_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),aa(num,num,bit0,one2))))),at_bot(real)) ).

% tendsto_arctan_at_bot
tff(fact_6828_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_6829_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_bot(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_6830_ln__at__0,axiom,
    filterlim(real,real,ln_ln(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% ln_at_0
tff(fact_6831_tendsto__at__botI__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [F2: fun(real,A),Y: A] :
          ( ! [X6: fun(nat,real)] :
              ( filterlim(nat,real,X6,at_bot(real),at_top(nat))
             => filterlim(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_ajg(fun(real,A),fun(fun(nat,real),fun(nat,A)),F2),X6),topolo7230453075368039082e_nhds(A,Y),at_top(nat)) )
         => filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,Y),at_bot(real)) ) ) ).

% tendsto_at_botI_sequentially
tff(fact_6832_filterlim__inverse__at__bot__neg,axiom,
    filterlim(real,real,inverse_inverse(real),at_bot(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_lessThan(real),zero_zero(real)))) ).

% filterlim_inverse_at_bot_neg
tff(fact_6833_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)),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,uminus_uminus(real),one_one(real))))) ).

% artanh_real_at_right_1
tff(fact_6834_at__bot__sub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_ajh(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C2))) ) ).

% at_bot_sub
tff(fact_6835_at__bot__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_aji(A,filter(A))),top_top(set(A)))) ) ) ).

% at_bot_def
tff(fact_6836_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
         => ? [Y5: real] :
              ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X2,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y5) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_6837_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ajj(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_6838_at__infinity__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( at_infinity(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(real),set(filter(A)),image(real,filter(A),aTP_Lamp_ajl(real,filter(A))),top_top(set(real)))) ) ) ).

% at_infinity_def
tff(fact_6839_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F2: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F2,at_bot(real),F4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ajj(nat,fun(fun(real,real),fun(real,real)),Nb),F2),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_6840_exp__at__top,axiom,
    filterlim(real,real,exp(real),at_top(real),at_top(real)) ).

% exp_at_top
tff(fact_6841_ln__at__top,axiom,
    filterlim(real,real,ln_ln(real),at_top(real),at_top(real)) ).

% ln_at_top
tff(fact_6842_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_6843_filterlim__at__top__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F4)
     => ( filterlim(A,real,G,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ).

% filterlim_at_top_add_at_top
tff(fact_6844_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,at_top(real),F4)
     => ( filterlim(A,real,G,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_6845_filterlim__at__infinity__imp__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => filterlim(A,real,aTP_Lamp_adb(fun(A,B),fun(A,real),F2),at_top(real),F4) ) ) ).

% filterlim_at_infinity_imp_norm_at_top
tff(fact_6846_filterlim__norm__at__top__imp__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_adb(fun(A,B),fun(A,real),F2),at_top(real),F4)
         => filterlim(A,B,F2,at_infinity(B),F4) ) ) ).

% filterlim_norm_at_top_imp_at_infinity
tff(fact_6847_filterlim__at__infinity__conv__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G7: filter(A)] :
          ( filterlim(A,B,F2,at_infinity(B),G7)
        <=> filterlim(A,real,aTP_Lamp_adb(fun(A,B),fun(A,real),F2),at_top(real),G7) ) ) ).

% filterlim_at_infinity_conv_norm_at_top
tff(fact_6848_filterlim__tendsto__add__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( filterlim(A,real,G,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajm(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ).

% filterlim_tendsto_add_at_top
tff(fact_6849_filterlim__real__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_top(real),at_top(nat)) ).

% filterlim_real_sequentially
tff(fact_6850_filterlim__uminus__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F4)
    <=> filterlim(A,real,aTP_Lamp_ajn(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ).

% filterlim_uminus_at_top
tff(fact_6851_filterlim__uminus__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_bot(real),F4)
    <=> filterlim(A,real,aTP_Lamp_ajn(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ).

% filterlim_uminus_at_bot
tff(fact_6852_filterlim__at__top__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_aig(fun(real,A),fun(real,A),F2),F4,at_bot(real)) ) ).

% filterlim_at_top_mirror
tff(fact_6853_filterlim__at__bot__mirror,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_bot(real))
    <=> filterlim(real,A,aTP_Lamp_aig(fun(real,A),fun(real,A),F2),F4,at_top(real)) ) ).

% filterlim_at_bot_mirror
tff(fact_6854_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_6855_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_6856_filterlim__pow__at__top,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aci(nat,fun(fun(A,real),fun(A,real)),Nb),F2),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_6857_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_6858_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_6859_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_6860_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( filterlim(A,real,G,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajp(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% real_tendsto_divide_at_top
tff(fact_6861_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_top(real),F4)
     => filterlim(A,real,aTP_Lamp_ajq(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ).

% tendsto_inverse_0_at_top
tff(fact_6862_filterlim__sequentially__iff__filterlim__real,axiom,
    ! [A: $tType,F2: fun(A,nat),F4: filter(A)] :
      ( filterlim(A,nat,F2,at_top(nat),F4)
    <=> filterlim(A,real,aTP_Lamp_ajr(fun(A,nat),fun(A,real),F2),at_top(real),F4) ) ).

% filterlim_sequentially_iff_filterlim_real
tff(fact_6863_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_6864_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajs(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_6865_tendsto__neg__powr,axiom,
    ! [A: $tType,S3: real,F2: fun(A,real),F4: filter(A)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),S3),zero_zero(real))
     => ( filterlim(A,real,F2,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajt(real,fun(fun(A,real),fun(A,real)),S3),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_neg_powr
tff(fact_6866_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( ( C2 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F4)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aju(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_6867_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_ajv(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_6868_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( filterlim(A,B,G,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajw(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_6869_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_ajx(fun(A,B),fun(nat,fun(A,B)),F2),Nb),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_6870_filterlim__at__top__to__right,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,at_top(real))
    <=> filterlim(real,A,aTP_Lamp_ajy(fun(real,A),fun(real,A),F2),F4,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% filterlim_at_top_to_right
tff(fact_6871_filterlim__at__right__to__top,axiom,
    ! [A: $tType,F2: fun(real,A),F4: filter(A)] :
      ( filterlim(real,A,F2,F4,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
    <=> filterlim(real,A,aTP_Lamp_ajy(fun(real,A),fun(real,A),F2),F4,at_top(real)) ) ).

% filterlim_at_right_to_top
tff(fact_6872_artanh__real__at__left__1,axiom,
    filterlim(real,real,artanh(real),at_top(real),topolo174197925503356063within(real,one_one(real),aa(real,set(real),set_ord_lessThan(real),one_one(real)))) ).

% artanh_real_at_left_1
tff(fact_6873_filterlim__inverse__at__top__right,axiom,
    filterlim(real,real,inverse_inverse(real),at_top(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% filterlim_inverse_at_top_right
tff(fact_6874_filterlim__inverse__at__right__top,axiom,
    filterlim(real,real,inverse_inverse(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))),at_top(real)) ).

% filterlim_inverse_at_right_top
tff(fact_6875_tendsto__at__topI__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [F2: fun(real,A),Y: A] :
          ( ! [X6: fun(nat,real)] :
              ( filterlim(nat,real,X6,at_top(real),at_top(nat))
             => filterlim(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_ajg(fun(real,A),fun(fun(nat,real),fun(nat,A)),F2),X6),topolo7230453075368039082e_nhds(A,Y),at_top(nat)) )
         => filterlim(real,A,F2,topolo7230453075368039082e_nhds(A,Y),at_top(real)) ) ) ).

% tendsto_at_topI_sequentially
tff(fact_6876_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),zero_zero(real))
       => ( filterlim(A,real,G,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajf(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_6877_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_ajz(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_6878_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_aka(fun(real,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% lim_infinity_imp_sequentially
tff(fact_6879_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_ade(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F4)
        <=> filterlim(A,B,G,at_infinity(B),F4) ) ) ).

% filterlim_inverse_at_iff
tff(fact_6880_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim(real,real,aTP_Lamp_akb(real,fun(real,real),Xb),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xb)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_6881_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C2: A,F4: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C2),F4)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_wm(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_6882_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun(real,real),Flim: real] :
      ( ! [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),X2)
         => ? [Y5: real] :
              ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X2,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),zero_zero(real)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_6883_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),aa(num,num,bit0,one2)))),at_top(real)) ).

% tendsto_arctan_at_top
tff(fact_6884_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xb))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xb),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_6885_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_akc(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_6886_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),aa(num,num,bit0,one2))),aa(real,set(real),set_ord_lessThan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% filterlim_tan_at_left
tff(fact_6887_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,Nb: nat,B4: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_akd(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C2),Nb),B4),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_6888_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),Xb: real,G5: fun(real,real),F2: fun(real,real),F6: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
     => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F6),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_6889_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_akh(fun(nat,$o),fun(nat,$o),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_6890_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_aki(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_6891_eventually__const,axiom,
    ! [A: $tType,F4: filter(A),P: $o] :
      ( ( F4 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_tc($o,fun(A,$o),(P)),F4)
      <=> (P) ) ) ).

% eventually_const
tff(fact_6892_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_pz(A,fun(A,$o)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_6893_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N5: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N4),N5)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_6894_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_akj(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_6895_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_akk(A,fun(A,$o),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_6896_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N5: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N4),N5)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_bot_dense
tff(fact_6897_eventually__not__equal__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : eventually(A,aTP_Lamp_akl(A,fun(A,$o),A2),at_infinity(A)) ) ).

% eventually_not_equal_at_infinity
tff(fact_6898_le__sequentially,axiom,
    ! [F4: filter(nat)] :
      ( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),F4),at_top(nat))
    <=> ! [N5: nat] : eventually(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),F4) ) ).

% le_sequentially
tff(fact_6899_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less_eq(A),C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_6900_eventually__sequentially,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N5: nat] :
        ! [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N4)
         => aa(nat,$o,P,N4) ) ) ).

% eventually_sequentially
tff(fact_6901_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,$o)] :
      ( ! [X2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),X2)
         => aa(nat,$o,P,X2) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_6902_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N5: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N5),N4)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_top_linorder
tff(fact_6903_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,$o)] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X2)
             => aa(A,$o,P,X2) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_6904_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_pg(nat,$o),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_6905_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_akm(A,fun(A,$o),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_6906_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_6907_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N5: A] :
            ! [N4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N5),N4)
             => aa(A,$o,P,N4) ) ) ) ).

% eventually_at_top_dense
tff(fact_6908_summable__cong,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_akn(fun(nat,A),fun(fun(nat,A),fun(nat,$o)),F2),G),at_top(nat))
         => ( summable(A,F2)
          <=> summable(A,G) ) ) ) ).

% summable_cong
tff(fact_6909_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_aki(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_6910_sequentially__imp__eventually__at,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [A2: A,P: fun(A,$o)] :
          ( ! [F3: fun(nat,A)] :
              ( ( ! [N8: nat] : aa(nat,A,F3,N8) != A2
                & filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ako(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% sequentially_imp_eventually_at
tff(fact_6911_sequentially__imp__eventually__within,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [S3: set(A),A2: A,P: fun(A,$o)] :
          ( ! [F3: fun(nat,A)] :
              ( ( ! [N8: nat] :
                    ( aa(set(A),$o,member(A,aa(nat,A,F3,N8)),S3)
                    & ( aa(nat,A,F3,N8) != A2 ) )
                & filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ako(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A2,S3)) ) ) ).

% sequentially_imp_eventually_within
tff(fact_6912_filterlim__at__infinity__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_infinity(B),F4)
         => eventually(A,aa(B,fun(A,$o),aTP_Lamp_akp(fun(A,B),fun(B,fun(A,$o)),F2),C2),F4) ) ) ).

% filterlim_at_infinity_imp_eventually_ne
tff(fact_6913_eventually__nhds__iff__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ! [F5: fun(nat,A)] :
              ( filterlim(nat,A,F5,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ako(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F5),at_top(nat)) ) ) ) ).

% eventually_nhds_iff_sequentially
tff(fact_6914_sequentially__imp__eventually__nhds__within,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [S3: set(A),A2: A,P: fun(A,$o)] :
          ( ! [F3: fun(nat,A)] :
              ( ( ! [N8: nat] : aa(set(A),$o,member(A,aa(nat,A,F3,N8)),S3)
                & filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ako(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) )
         => eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A2)),aa(set(A),filter(A),principal(A),S3))) ) ) ).

% sequentially_imp_eventually_nhds_within
tff(fact_6915_eventually__nhds__within__iff__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [P: fun(A,$o),A2: A,S3: set(A)] :
          ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A2)),aa(set(A),filter(A),principal(A),S3)))
        <=> ! [F5: fun(nat,A)] :
              ( ( ! [N4: nat] : aa(set(A),$o,member(A,aa(nat,A,F5,N4)),S3)
                & filterlim(nat,A,F5,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ako(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F5),at_top(nat)) ) ) ) ).

% eventually_nhds_within_iff_sequentially
tff(fact_6916_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F4: filter(B),G7: filter(A),F7: filter(B),G8: filter(A),F6: fun(A,B)] :
      ( filterlim(A,B,F2,F4,G7)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F4),F7)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G8),G7)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_akq(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),F6),G8)
           => filterlim(A,B,F6,F7,G8) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_6917_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F4: filter(A),F2: fun(B,A),G7: filter(B)] :
      ( eventually(A,P,F4)
     => ( filterlim(B,A,F2,F4,G7)
       => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_akr(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2),G7) ) ) ).

% eventually_compose_filterlim
tff(fact_6918_filterlim__principal,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),S: set(B),F4: filter(A)] :
      ( filterlim(A,B,F2,aa(set(B),filter(B),principal(B),S),F4)
    <=> eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(set(B),fun(A,$o)),F2),S),F4) ) ).

% filterlim_principal
tff(fact_6919_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F12: filter(A),F13: filter(A),F23: filter(B),F24: filter(B),F2: fun(B,A),G: fun(B,A)] :
      ( ( F12 = F13 )
     => ( ( F23 = F24 )
       => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_akt(fun(B,A),fun(fun(B,A),fun(B,$o)),F2),G),F23)
         => ( filterlim(B,A,F2,F12,F23)
          <=> filterlim(B,A,G,F13,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_6920_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F23: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,F23,F12)
    <=> ! [P5: fun(B,$o)] :
          ( eventually(B,P5,F23)
         => eventually(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_aku(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P5),F12) ) ) ).

% filterlim_iff
tff(fact_6921_tendsto__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,Net: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_akv(fun(A,B),fun(B,fun(A,$o)),F2),L),Net)
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),Net) ) ) ).

% tendsto_eventually
tff(fact_6922_tendsto__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t1_space(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),C8: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( ( C2 != C8 )
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_akw(fun(A,B),fun(B,fun(A,$o)),F2),C8),F4) ) ) ) ).

% tendsto_imp_eventually_ne
tff(fact_6923_tendsto__discrete,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
        <=> eventually(A,aa(B,fun(A,$o),aTP_Lamp_akx(fun(A,B),fun(B,fun(A,$o)),F2),Y),F4) ) ) ).

% tendsto_discrete
tff(fact_6924_tendsto__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),G: fun(A,B),F4: filter(A),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aky(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F4) ) ) ) ).

% tendsto_cong
tff(fact_6925_Lim__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aky(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% Lim_transform_eventually
tff(fact_6926_filterlim__at__within__not__equal,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(B)
     => ! [F2: fun(A,B),A2: B,S3: set(B),F4: filter(A),B2: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,A2,S3),F4)
         => eventually(A,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_akz(fun(A,B),fun(set(B),fun(B,fun(A,$o))),F2),S3),B2),F4) ) ) ).

% filterlim_at_within_not_equal
tff(fact_6927_le__principal,axiom,
    ! [A: $tType,F4: filter(A),A3: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),aa(set(A),filter(A),principal(A),A3))
    <=> eventually(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3),F4) ) ).

% le_principal
tff(fact_6928_eventually__inf__principal,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),S3: set(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),aa(set(A),filter(A),principal(A),S3)))
    <=> eventually(A,aa(set(A),fun(A,$o),aTP_Lamp_ala(fun(A,$o),fun(set(A),fun(A,$o)),P),S3),F4) ) ).

% eventually_inf_principal
tff(fact_6929_eventually__nhds__in__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),Xb: A] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),S3)
         => ( aa(set(A),$o,member(A,Xb),S3)
           => eventually(A,aTP_Lamp_alb(set(A),fun(A,$o),S3),topolo7230453075368039082e_nhds(A,Xb)) ) ) ) ).

% eventually_nhds_in_open
tff(fact_6930_eventually__eventually,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,aTP_Lamp_alc(fun(A,$o),fun(A,$o),P),topolo7230453075368039082e_nhds(A,Xb))
        <=> eventually(A,P,topolo7230453075368039082e_nhds(A,Xb)) ) ) ).

% eventually_eventually
tff(fact_6931_t1__space__nhds,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => eventually(A,aTP_Lamp_ald(A,fun(A,$o),Y),topolo7230453075368039082e_nhds(A,Xb)) ) ) ).

% t1_space_nhds
tff(fact_6932_eventually__at__filter,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),A2: A,S3: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S3))
        <=> eventually(A,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_ale(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),P),A2),S3),topolo7230453075368039082e_nhds(A,A2)) ) ) ).

% eventually_at_filter
tff(fact_6933_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( ~ eventually(A,Q,F4)
       => ~ eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_alf(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4) ) ) ).

% not_eventually_impI
tff(fact_6934_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
    <=> ( eventually(A,P,F4)
        & eventually(A,Q,F4) ) ) ).

% eventually_conj_iff
tff(fact_6935_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_alf(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
       => eventually(A,Q,F4) ) ) ).

% eventually_rev_mp
tff(fact_6936_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_alg(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
     => ( eventually(A,P,F4)
      <=> eventually(A,Q,F4) ) ) ).

% eventually_subst
tff(fact_6937_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o),R2: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,Q,F4)
       => ( ! [I3: A] :
              ( aa(A,$o,P,I3)
             => ( aa(A,$o,Q,I3)
               => aa(A,$o,R2,I3) ) )
         => eventually(A,R2,F4) ) ) ) ).

% eventually_elim2
tff(fact_6938_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,$o),F4: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F4)
     => ( eventually(A,Q,F4)
       => eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4) ) ) ).

% eventually_conj
tff(fact_6939_eventually__True,axiom,
    ! [A: $tType,F4: filter(A)] : eventually(A,aTP_Lamp_tn(A,$o),F4) ).

% eventually_True
tff(fact_6940_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_alf(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F4)
     => ( eventually(A,P,F4)
       => eventually(A,Q,F4) ) ) ).

% eventually_mp
tff(fact_6941_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,$o),C5: $o,F4: filter(A)] :
      ( eventually(A,aa($o,fun(A,$o),aTP_Lamp_alh(fun(A,$o),fun($o,fun(A,$o)),P),(C5)),F4)
    <=> ( eventually(A,P,F4)
        | (C5) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_6942_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C5: $o,P: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ali($o,fun(fun(A,$o),fun(A,$o)),(C5)),P),F4)
    <=> ( (C5)
        | eventually(A,P,F4) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_6943_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C5: $o,P: fun(A,$o),F4: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_alj($o,fun(fun(A,$o),fun(A,$o)),(C5)),P),F4)
    <=> ( (C5)
       => eventually(A,P,F4) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_6944_has__field__derivative__cong__eventually,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),G: fun(A,A),Xb: A,S: set(A),U: A] :
          ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_alk(fun(A,A),fun(fun(A,A),fun(A,$o)),F2),G),topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,F2,Xb) = aa(A,A,G,Xb) )
           => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,Xb,S))
            <=> has_field_derivative(A,G,U,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_field_derivative_cong_eventually
tff(fact_6945_trivial__limit__def,axiom,
    ! [A: $tType,F4: filter(A)] :
      ( ( F4 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_ag(A,$o),F4) ) ).

% trivial_limit_def
tff(fact_6946_eventually__const__iff,axiom,
    ! [A: $tType,P: $o,F4: filter(A)] :
      ( eventually(A,aTP_Lamp_tc($o,fun(A,$o),(P)),F4)
    <=> ( (P)
        | ( F4 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_6947_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( ! [X2: A] : ~ aa(A,$o,P,X2)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_6948_has__derivative__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),G: fun(A,B)] :
          ( has_derivative(A,B,F2,F6,topolo174197925503356063within(A,Xb,S3))
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_all(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,F2,Xb) = aa(A,B,G,Xb) )
             => ( aa(set(A),$o,member(A,Xb),S3)
               => has_derivative(A,B,G,F6,topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ) ).

% has_derivative_transform_eventually
tff(fact_6949_eventually__Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [P: fun(A,fun(A,$o)),Xb: A,X5: set(A)] :
          ( eventually(A,aa(A,fun(A,$o),P,topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,Xb,X5),aTP_Lamp_aac(A,A))),topolo174197925503356063within(A,Xb,X5))
        <=> eventually(A,aa(A,fun(A,$o),P,Xb),topolo174197925503356063within(A,Xb,X5)) ) ) ).

% eventually_Lim_ident_at
tff(fact_6950_eventually__INF1,axiom,
    ! [B: $tType,A: $tType,I2: A,I5: set(A),P: fun(B,$o),F4: fun(A,filter(B))] :
      ( aa(set(A),$o,member(A,I2),I5)
     => ( eventually(B,P,aa(A,filter(B),F4,I2))
       => eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),I5))) ) ) ).

% eventually_INF1
tff(fact_6951_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),top_top(A))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),top_top(A))
                & ! [Z3: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Z3)
                   => aa(A,$o,P,Z3) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_6952_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A)] :
          ( ! [X2: A,Y4: A] :
              ( aa(A,$o,Q,X2)
             => ( aa(A,$o,Q,Y4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) ) ) )
         => ( ! [X2: B] :
                ( aa(B,$o,P,X2)
               => ( aa(A,B,F2,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( aa(B,$o,P,X2)
                 => aa(A,$o,Q,aa(B,A,G,X2)) )
             => ( 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_6953_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_lessThan(A),Xb)))
        <=> ? [B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Xb)
              & ! [Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb)
                   => aa(A,$o,P,Y3) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_6954_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xb: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( eventually(A,P,topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_lessThan(A),Xb)))
          <=> ? [B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Xb)
                & ! [Y3: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Y3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb)
                     => aa(A,$o,P,Y3) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_6955_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_greaterThan(A),Xb)))
        <=> ? [B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B5)
              & ! [Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),B5)
                   => aa(A,$o,P,Y3) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_6956_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,Y: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( eventually(A,P,topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_greaterThan(A),Xb)))
          <=> ? [B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B5)
                & ! [Y3: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),B5)
                     => aa(A,$o,P,Y3) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_6957_eventually__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_infinity(A))
        <=> ? [B5: real] :
            ! [X: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B5),real_V7770717601297561774m_norm(A,X))
             => aa(A,$o,P,X) ) ) ) ).

% eventually_at_infinity
tff(fact_6958_tendsto__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),G: fun(A,B),Net: filter(A),H: fun(A,B),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_alm(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_alm(fun(A,B),fun(fun(A,B),fun(A,$o)),G),H),Net)
           => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),Net)
             => ( filterlim(A,B,H,topolo7230453075368039082e_nhds(B,C2),Net)
               => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_6959_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),A2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aln(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_6960_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_alo(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_6961_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F2: fun(B,A),F4: filter(B)] :
          ( ! [A4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_alp(fun(B,A),fun(A,fun(B,$o)),F2),A4),F4) )
         => ( ! [A4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A4)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_alq(fun(B,A),fun(A,fun(B,$o)),F2),A4),F4) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Y),F4) ) ) ) ).

% order_tendstoI
tff(fact_6962_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
        <=> ( ! [L4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L4),Xb)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_alo(fun(A,B),fun(B,fun(A,$o)),F2),L4),F4) )
            & ! [U4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xb),U4)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aln(fun(A,B),fun(B,fun(A,$o)),F2),U4),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_6963_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_alr(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_top
tff(fact_6964_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C2),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_alr(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_6965_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,at_top(B),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_als(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
           => filterlim(A,B,G,at_top(B),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_6966_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_alt(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_6967_filterlim__at,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),B2: B,S3: set(B),F4: filter(A)] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,B2,S3),F4)
        <=> ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_alu(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),B2),S3),F4)
            & filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),F4) ) ) ) ).

% filterlim_at
tff(fact_6968_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),Xb),topolo174197925503356063within(A,Xb,aa(A,set(A),set_ord_greaterThan(A),Xb))) ) ).

% eventually_at_right_less
tff(fact_6969_has__field__derivative__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,Y: A,S: set(A),F2: fun(A,A),G: fun(A,A),U: A,V: A,Ta: set(A)] :
          ( ( Xb = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_alv(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),S),F2),G),topolo7230453075368039082e_nhds(A,Xb))
           => ( ( U = V )
             => ( ( S = Ta )
               => ( aa(set(A),$o,member(A,Xb),S)
                 => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,Xb,S))
                  <=> has_field_derivative(A,G,V,topolo174197925503356063within(A,Y,Ta)) ) ) ) ) ) ) ) ).

% has_field_derivative_cong_ev
tff(fact_6970_tendsto__def,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [S9: set(B)] :
              ( aa(set(B),$o,topolo1002775350975398744n_open(B),S9)
             => ( aa(set(B),$o,member(B,L),S9)
               => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_alw(fun(A,B),fun(set(B),fun(A,$o)),F2),S9),F4) ) ) ) ) ).

% tendsto_def
tff(fact_6971_topological__tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),S: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( aa(set(B),$o,topolo1002775350975398744n_open(B),S)
           => ( aa(set(B),$o,member(B,L),S)
             => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_alw(fun(A,B),fun(set(B),fun(A,$o)),F2),S),F4) ) ) ) ) ).

% topological_tendstoD
tff(fact_6972_topological__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [L: A,F2: fun(B,A),F4: filter(B)] :
          ( ! [S5: set(A)] :
              ( aa(set(A),$o,topolo1002775350975398744n_open(A),S5)
             => ( aa(set(A),$o,member(A,L),S5)
               => eventually(B,aa(set(A),fun(B,$o),aTP_Lamp_alx(fun(B,A),fun(set(A),fun(B,$o)),F2),S5),F4) ) )
         => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4) ) ) ).

% topological_tendstoI
tff(fact_6973_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z6),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aly(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_6974_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_aly(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_bot
tff(fact_6975_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_alz(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_6976_real__tendsto__sandwich,axiom,
    ! [A: $tType,F2: fun(A,real),G: fun(A,real),Net: filter(A),H: fun(A,real),C2: real] :
      ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_ama(fun(A,real),fun(fun(A,real),fun(A,$o)),F2),G),Net)
     => ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_ama(fun(A,real),fun(fun(A,real),fun(A,$o)),G),H),Net)
       => ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C2),Net)
         => ( filterlim(A,real,H,topolo7230453075368039082e_nhds(real,C2),Net)
           => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,C2),Net) ) ) ) ) ).

% real_tendsto_sandwich
tff(fact_6977_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Xb: A] :
          ~ ! [A7: fun(nat,set(A))] :
              ( ! [I: nat] : aa(set(A),$o,topolo1002775350975398744n_open(A),aa(nat,set(A),A7,I))
             => ( ! [I: nat] : aa(set(A),$o,member(A,Xb),aa(nat,set(A),A7,I))
               => ~ ! [S10: set(A)] :
                      ( aa(set(A),$o,topolo1002775350975398744n_open(A),S10)
                     => ( aa(set(A),$o,member(A,Xb),S10)
                       => eventually(nat,aa(set(A),fun(nat,$o),aTP_Lamp_amb(fun(nat,set(A)),fun(set(A),fun(nat,$o)),A7),S10),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_6978_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,$o),F4: fun(A,filter(B))] :
      ( finite_finite2(A,A3)
     => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),A3)))
      <=> ? [Q7: fun(A,fun(B,$o))] :
            ( ! [X: A] :
                ( aa(set(A),$o,member(A,X),A3)
               => eventually(B,aa(A,fun(B,$o),Q7,X),aa(A,filter(B),F4,X)) )
            & ! [Y3: B] :
                ( ! [X: A] :
                    ( aa(set(A),$o,member(A,X),A3)
                   => aa(B,$o,aa(A,fun(B,$o),Q7,X),Y3) )
               => aa(B,$o,P,Y3) ) ) ) ) ).

% eventually_INF_finite
tff(fact_6979_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S))
        <=> ? [D4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
              & ! [X: A] :
                  ( aa(set(A),$o,member(A,X),S)
                 => ( ( ( X != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),D4) )
                   => aa(A,$o,P,X) ) ) ) ) ) ).

% eventually_at
tff(fact_6980_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
              & ! [X: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),D4)
                 => aa(A,$o,P,X) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_6981_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_6982_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% eventually_at_rightI
tff(fact_6983_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o),A2: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_amc(fun(A,$o),fun(A,fun(A,$o)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_6984_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_amd(fun(A,B),fun(B,fun(A,$o)),F2),L),F4)
         => ( ! [X2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X2),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_alo(fun(A,B),fun(B,fun(A,$o)),F2),X2),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_6985_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F2: fun(A,B),F4: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ame(B,fun(fun(A,B),fun(A,$o)),L),F2),F4)
         => ( ! [X2: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X2)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aln(fun(A,B),fun(B,fun(A,$o)),F2),X2),F4) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_6986_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C2),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_amf(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_6987_tendsto__compose__eventually,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),Mb: B,L: A,F2: fun(C,A),F4: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Mb),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
           => ( eventually(C,aa(fun(C,A),fun(C,$o),aTP_Lamp_amg(A,fun(fun(C,A),fun(C,$o)),L),F2),F4)
             => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aex(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),topolo7230453075368039082e_nhds(B,Mb),F4) ) ) ) ) ).

% tendsto_compose_eventually
tff(fact_6988_LIM__compose__eventually,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_amh(fun(A,B),fun(B,fun(A,$o)),F2),B2),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ami(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_6989_filterlim__atI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_amj(fun(A,B),fun(B,fun(A,$o)),F2),C2),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C2,top_top(set(B))),F4) ) ) ) ).

% filterlim_atI
tff(fact_6990_isCont__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),G: fun(A,B),Xb: A] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_amk(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),topolo7230453075368039082e_nhds(A,Xb))
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),F2)
          <=> topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),G) ) ) ) ).

% isCont_cong
tff(fact_6991_DERIV__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,Y: A,F2: fun(A,A),G: fun(A,A),U: A,V: A] :
          ( ( Xb = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_alk(fun(A,A),fun(fun(A,A),fun(A,$o)),F2),G),topolo7230453075368039082e_nhds(A,Xb))
           => ( ( U = V )
             => ( has_field_derivative(A,F2,U,topolo174197925503356063within(A,Xb,top_top(set(A))))
              <=> has_field_derivative(A,G,V,topolo174197925503356063within(A,Y,top_top(set(A)))) ) ) ) ) ) ).

% DERIV_cong_ev
tff(fact_6992_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F4: filter(A),C2: B] :
          ( filterlim(A,B,F2,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z6),C2)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aml(fun(A,B),fun(B,fun(A,$o)),F2),Z6),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_6993_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_amm(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),A2) ) ) ) ) ).

% tendsto_upperbound
tff(fact_6994_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),Xb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_amn(fun(A,B),fun(B,fun(A,$o)),F2),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),Xb) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_6995_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F4: filter(A),F2: fun(A,B),Xb: B,G: fun(A,B),Y: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xb),F4)
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Y),F4)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_amo(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G),F4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Xb) ) ) ) ) ) ).

% tendsto_le
tff(fact_6996_metric__tendsto__imp__tendsto,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),A2: B,F4: filter(A),G: fun(A,C),B2: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( eventually(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_amp(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),F2),A2),G),B2),F4)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B2),F4) ) ) ) ).

% metric_tendsto_imp_tendsto
tff(fact_6997_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_amq(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,F2,at_top(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_6998_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_amr(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,F2,at_bot(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_6999_eventually__floor__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ams(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_floor_eq
tff(fact_7000_eventually__ceiling__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_amt(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_ceiling_eq
tff(fact_7001_eventually__at__right__to__0,axiom,
    ! [P: fun(real,$o),A2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
    <=> eventually(real,aa(real,fun(real,$o),aTP_Lamp_amu(fun(real,$o),fun(real,fun(real,$o)),P),A2),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% eventually_at_right_to_0
tff(fact_7002_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F4: fun(B,filter(A)),B4: set(B)] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),F4),B4)))
    <=> ? [X7: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X7),B4)
          & finite_finite2(B,X7)
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),F4),X7))) ) ) ).

% eventually_INF
tff(fact_7003_eventually__at__left__to__right,axiom,
    ! [P: fun(real,$o),A2: real] :
      ( eventually(real,P,topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
    <=> eventually(real,aTP_Lamp_amv(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,uminus_uminus(real),A2)))) ) ).

% eventually_at_left_to_right
tff(fact_7004_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F2)
         => ( eventually(A,aTP_Lamp_amw(fun(A,real),fun(A,$o),F2),F4)
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_aib(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_7005_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_amx(real,fun(real,fun(real,$o)),A2),B2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ).

% eventually_at_right_real
tff(fact_7006_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B2),A2)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_amx(real,fun(real,fun(real,$o)),B2),A2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2))) ) ).

% eventually_at_left_real
tff(fact_7007_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A2: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S))
        <=> ? [D4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
              & ! [X: A] :
                  ( aa(set(A),$o,member(A,X),S)
                 => ( ( ( X != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,A2)),D4) )
                   => aa(A,$o,P,X) ) ) ) ) ) ).

% eventually_at_le
tff(fact_7008_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_infinity(A))
        <=> ? [B5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B5)
              & ! [X: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B5),real_V7770717601297561774m_norm(A,X))
                 => aa(A,$o,P2,X) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_7009_tendsto__compose__at,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),Y: B,F4: filter(A),G: fun(B,C),Z: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,Z),topolo174197925503356063within(B,Y,top_top(set(B))))
           => ( eventually(A,aa(C,fun(A,$o),aa(fun(B,C),fun(C,fun(A,$o)),aa(B,fun(fun(B,C),fun(C,fun(A,$o))),aTP_Lamp_amy(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),F2),Y),G),Z),F4)
             => filterlim(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,G),F2),topolo7230453075368039082e_nhds(C,Z),F4) ) ) ) ) ).

% tendsto_compose_at
tff(fact_7010_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_aln(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_lessThan(B),L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_7011_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_alo(fun(A,B),fun(B,fun(A,$o)),F2),L5),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_greaterThan(B),L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_7012_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_amz(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E4),F4) ) ) ) ).

% tendsto_iff
tff(fact_7013_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_amz(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E2),F4) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% tendstoI
tff(fact_7014_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A),E: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
           => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_amz(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E),F4) ) ) ) ).

% tendstoD
tff(fact_7015_summable__comparison__test__ev,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ana(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G),at_top(nat))
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test_ev
tff(fact_7016_eventually__at__top__to__right,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,at_top(real))
    <=> eventually(real,aTP_Lamp_anb(fun(real,$o),fun(real,$o),P),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% eventually_at_top_to_right
tff(fact_7017_eventually__at__right__to__top,axiom,
    ! [P: fun(real,$o)] :
      ( eventually(real,P,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
    <=> eventually(real,aTP_Lamp_anb(fun(real,$o),fun(real,$o),P),at_top(real)) ) ).

% eventually_at_right_to_top
tff(fact_7018_tendsto__arcosh__strong,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),A2)
       => ( eventually(A,aTP_Lamp_anc(fun(A,real),fun(A,$o),F2),F4)
         => filterlim(A,real,aTP_Lamp_acz(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_7019_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A2: A] :
          ( ! [X2: A,Y4: A] :
              ( aa(A,$o,Q,X2)
             => ( aa(A,$o,Q,Y4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) ) ) )
         => ( ! [X2: B] :
                ( aa(B,$o,P,X2)
               => ( aa(A,B,F2,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( aa(B,$o,P,X2)
                 => aa(A,$o,Q,aa(B,A,G,X2)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2)))
               => ( ! [B3: A] :
                      ( aa(A,$o,Q,B3)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A2) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_7020_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B4: set(A),F4: fun(A,filter(B)),P: fun(B,$o)] :
      ( ( B4 != bot_bot(set(A)) )
     => ( ! [A4: A] :
            ( aa(set(A),$o,member(A,A4),B4)
           => ! [B3: A] :
                ( aa(set(A),$o,member(A,B3),B4)
               => ? [X4: A] :
                    ( aa(set(A),$o,member(A,X4),B4)
                    & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,A4)),aa(A,filter(B),F4,B3))) ) ) )
       => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),B4)))
        <=> ? [X: A] :
              ( aa(set(A),$o,member(A,X),B4)
              & eventually(B,P,aa(A,filter(B),F4,X)) ) ) ) ) ).

% eventually_INF_base
tff(fact_7021_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A2: A] :
          ( ! [X2: A,Y4: A] :
              ( aa(A,$o,Q,X2)
             => ( aa(A,$o,Q,Y4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) ) ) )
         => ( ! [X2: B] :
                ( aa(B,$o,P,X2)
               => ( aa(A,B,F2,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( aa(B,$o,P,X2)
                 => aa(A,$o,Q,aa(B,A,G,X2)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
               => ( ! [B3: A] :
                      ( aa(A,$o,Q,B3)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B3) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_7022_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F4: filter(A),G: fun(A,C),K5: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_and(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F2),G),K5),F4)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_7023_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C2: B,F4: filter(A),A3: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C2),F4)
         => ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ane(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),C2),A3),F4)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C2,A3),F4) ) ) ) ).

% filterlim_at_withinI
tff(fact_7024_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C2: real,F2: fun(A,B),F4: filter(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
         => ( filterlim(A,B,F2,at_infinity(B),F4)
          <=> ! [R5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),R5)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_anf(fun(A,B),fun(real,fun(A,$o)),F2),R5),F4) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_7025_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_ang(fun(A,real),fun(A,$o),F2),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_7026_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_ang(fun(A,real),fun(A,$o),F2),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ) ).

% tendsto_powr2
tff(fact_7027_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( ( ( A2 != zero_zero(real) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
              & eventually(A,aTP_Lamp_ang(fun(A,real),fun(A,$o),F2),F4) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_adc(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ).

% tendsto_powr'
tff(fact_7028_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_anh(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_7029_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ aa(set(B),$o,member(B,L),ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ani(fun(A,B),fun(B,fun(A,$o)),F2),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_7030_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A2: real,F4: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => ( eventually(A,aTP_Lamp_amq(fun(A,real),fun(A,$o),G),F4)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajp(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F4) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_7031_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_amq(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,F2,at_top(real),F4)
      <=> filterlim(A,real,aa(fun(A,real),fun(A,real),comp(real,real,A,inverse_inverse(real)),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_7032_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_amq(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_ajq(fun(A,real),fun(A,real),F2),at_top(real),F4) ) ) ).

% filterlim_inverse_at_top
tff(fact_7033_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_amq(fun(A,real),fun(A,$o),F2),F4)
     => ( filterlim(A,real,aTP_Lamp_ajq(fun(A,real),fun(A,real),F2),at_top(real),F4)
      <=> filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_7034_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_amr(fun(A,real),fun(A,$o),F2),F4)
       => filterlim(A,real,aTP_Lamp_ajq(fun(A,real),fun(A,real),F2),at_bot(real),F4) ) ) ).

% filterlim_inverse_at_bot
tff(fact_7035_lhopital__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G5: 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,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G5),at_top(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(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_7036_lhopital,axiom,
    ! [F2: fun(real,real),Xb: real,G: fun(real,real),G5: fun(real,real),F6: fun(real,real),F4: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,top_top(set(real))))
       => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,Xb,top_top(set(real))))
         => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,top_top(set(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,Xb,top_top(set(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,top_top(set(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F6),F4,topolo174197925503356063within(real,Xb,top_top(set(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F4,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_7037_lhopital__right__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G5: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G5),at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_7038_lhopital__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G5: 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,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,A2,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G5),at_bot(real),topolo174197925503356063within(real,A2,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(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_7039_lhopital__left__at__top__at__top,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G5: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G5),at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_7040_lhopital__at__top,axiom,
    ! [G: fun(real,real),Xb: real,G5: fun(real,real),F2: fun(real,real),F6: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,Xb,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F6),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_7041_lhospital__at__top__at__top,axiom,
    ! [G: fun(real,real),G5: fun(real,real),F2: fun(real,real),F6: fun(real,real),Xb: real] :
      ( filterlim(real,real,G,at_top(real),at_top(real))
     => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G5),at_top(real))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),at_top(real))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),at_top(real))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F6),topolo7230453075368039082e_nhds(real,Xb),at_top(real))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Xb),at_top(real)) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_7042_lhopital__right__0,axiom,
    ! [F0: fun(real,real),G0: fun(real,real),G5: fun(real,real),F6: fun(real,real),F4: filter(real)] :
      ( filterlim(real,real,F0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
     => ( filterlim(real,real,G0,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
       => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G0),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
         => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F0),F6),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G0),G5),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F6),F4,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F0),G0),F4,topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_7043_lhopital__right,axiom,
    ! [F2: fun(real,real),Xb: real,G: fun(real,real),G5: fun(real,real),F6: fun(real,real),F4: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
       => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
         => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F6),F4,topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F4,topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_7044_lhopital__left,axiom,
    ! [F2: fun(real,real),Xb: real,G: fun(real,real),G5: fun(real,real),F6: fun(real,real),F4: filter(real)] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
       => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
         => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F6),F4,topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),F4,topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_lessThan(real),Xb))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_7045_lhopital__right__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G5: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G5),at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_7046_lhopital__left__at__top__at__bot,axiom,
    ! [F2: fun(real,real),A2: real,G: fun(real,real),F6: fun(real,real),G5: fun(real,real)] :
      ( filterlim(real,real,F2,at_top(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
     => ( filterlim(real,real,G,at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F6),G5),at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(fun(real,real),fun(fun(real,real),fun(real,real)),F2),G),at_bot(real),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_7047_lhopital__right__0__at__top,axiom,
    ! [G: fun(real,real),G5: fun(real,real),F2: fun(real,real),F6: fun(real,real),Xb: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
     => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F6),topolo7230453075368039082e_nhds(real,Xb),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Xb),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_7048_lhopital__right__at__top,axiom,
    ! [G: fun(real,real),Xb: real,G5: fun(real,real),F2: fun(real,real),F6: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
     => ( eventually(real,aTP_Lamp_ake(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),F2),F6),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F6),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(fun(real,real),fun(fun(real,real),fun(real,real)),G),F2),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,aa(real,set(real),set_ord_greaterThan(real),Xb))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_7049_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ank(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),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_7050_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [Y3: B,K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_anl(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),Y3),K6),F4) ) ) ) ).

% Bfun_metric_def
tff(fact_7051_eventually__all__finite,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(A)
     => ! [P: fun(B,fun(A,$o)),Net: filter(B)] :
          ( ! [Y4: A] : eventually(B,aa(A,fun(B,$o),aTP_Lamp_anm(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y4),Net)
         => eventually(B,aTP_Lamp_ann(fun(B,fun(A,$o)),fun(B,$o),P),Net) ) ) ).

% eventually_all_finite
tff(fact_7052_Bfun__const,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [C2: B,F4: filter(A)] : bfun(A,B,aTP_Lamp_ano(B,fun(A,B),C2),F4) ) ).

% Bfun_const
tff(fact_7053_Bseq__offset,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),K: nat] :
          ( bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_anp(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat))
         => bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_7054_Bseq__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X5: fun(nat,A),K: nat] :
          ( bfun(nat,A,X5,at_top(nat))
         => bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_anp(fun(nat,A),fun(nat,fun(nat,A)),X5),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_7055_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_anq(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F2),G),at_top(nat)) ) ) ).

% Bseq_subseq
tff(fact_7056_Bseq__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_fb(fun(nat,A),fun(nat,A),X5),at_top(nat))
        <=> bfun(nat,A,X5,at_top(nat)) ) ) ).

% Bseq_minus_iff
tff(fact_7057_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_fa(fun(nat,A),fun(nat,A),F2),at_top(nat))
        <=> bfun(nat,A,F2,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_7058_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_anr(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_7059_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_anr(fun(nat,A),fun(A,fun(nat,A)),F2),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_7060_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_ans(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F2),G),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_7061_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_ant(fun(A,$o),fun(A,$o),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_7062_BseqI_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),K5: real] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),K5)
         => bfun(nat,A,X5,at_top(nat)) ) ) ).

% BseqI'
tff(fact_7063_Collect__all__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_anu(fun(A,fun(B,$o)),fun(A,$o),P)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_anw(fun(A,fun(B,$o)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_all_eq
tff(fact_7064_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),D2: B] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(B,B4)
       => finite_finite2(fun(A,B),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_anx(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A3),B4),D2))) ) ) ).

% finite_set_of_finite_funs
tff(fact_7065_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_eo(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_7066_Bseq__eventually__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(nat,A),G: fun(nat,B)] :
          ( eventually(nat,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_any(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),F2),G),at_top(nat))
         => ( bfun(nat,B,G,at_top(nat))
           => bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_eventually_mono
tff(fact_7067_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,F2: fun(int,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_anz(fun(int,A),fun(nat,A),F2),F4,at_top(nat))
     => filterlim(int,A,F2,F4,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_7068_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
         => ? [K9: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K9)
              & ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N8))),K9) ) ) ) ).

% BseqD
tff(fact_7069_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
         => ~ ! [K9: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K9)
               => ~ ! [N8: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N8))),K9) ) ) ) ).

% BseqE
tff(fact_7070_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K5: real,X5: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
         => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N))),K5)
           => bfun(nat,A,X5,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_7071_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N4))),K6) ) ) ) ).

% Bseq_def
tff(fact_7072_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [N5: nat] :
            ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N5))) ) ) ).

% Bseq_iff1a
tff(fact_7073_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [N5: nat] :
            ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X5,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N5))) ) ) ).

% Bseq_iff
tff(fact_7074_Bseq__realpow,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => bfun(nat,real,aa(real,fun(nat,real),power_power(real),Xb),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_7075_BfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),K5: real,F4: filter(A)] :
          ( eventually(A,aa(real,fun(A,$o),aTP_Lamp_aoa(fun(A,B),fun(real,fun(A,$o)),F2),K5),F4)
         => bfun(A,B,F2,F4) ) ) ).

% BfunI
tff(fact_7076_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [X: A] :
                ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X5,N4)),aa(A,A,uminus_uminus(A),X)))),K3) ) ) ) ).

% Bseq_iff2
tff(fact_7077_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [N5: nat] :
                ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X5,N4)),aa(A,A,uminus_uminus(A),aa(nat,A,X5,N5))))),K3) ) ) ) ).

% Bseq_iff3
tff(fact_7078_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_ade(fun(A,B),fun(A,B),F2),F4) ) ) ) ).

% Bfun_inverse
tff(fact_7079_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
        <=> ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & eventually(A,aa(real,fun(A,$o),aTP_Lamp_aoa(fun(A,B),fun(real,fun(A,$o)),F2),K6),F4) ) ) ) ).

% Bfun_def
tff(fact_7080_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F2,F4)
         => ~ ! [B8: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B8)
               => ~ eventually(A,aa(real,fun(A,$o),aTP_Lamp_aoa(fun(A,B),fun(real,fun(A,$o)),F2),B8),F4) ) ) ) ).

% BfunE
tff(fact_7081_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aob(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),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_7082_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_aoc(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_7083_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( aa(set(A),$o,member(A,I2),set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I2),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_7084_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_7085_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_7086_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_7087_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ finite_finite2(A,set_or3652927894154168847AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioc_iff
tff(fact_7088_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_7089_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Xb,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_7090_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,Xb)) = Xb ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_7091_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Xb,Y)) = Xb ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_7092_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,Xb)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_7093_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or3652927894154168847AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).

% card_greaterThanAtMost
tff(fact_7094_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or7035219750837199246ssThan(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ).

% image_diff_atLeastLessThan
tff(fact_7095_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or3652927894154168847AtMost(A,A2,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_7096_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C2,D2) )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2) )
            | ( ( A2 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_7097_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_7098_GreatestI__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y4: nat] :
            ( aa(nat,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_nat
tff(fact_7099_Greatest__le__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y4: nat] :
            ( aa(nat,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),order_Greatest(nat,P)) ) ) ).

% Greatest_le_nat
tff(fact_7100_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),B2: nat] :
      ( ? [X_12: nat] : aa(nat,$o,P,X_12)
     => ( ! [Y4: nat] :
            ( aa(nat,$o,P,Y4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_ex_nat
tff(fact_7101_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ finite_finite2(A,set_or3652927894154168847AtMost(A,A2,B2)) ) ) ).

% infinite_Ioc
tff(fact_7102_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_7103_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),A2) ) ) ) ).

% Ioc_disjoint
tff(fact_7104_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),Xb: A,Y: A] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),S)
         => ( aa(set(A),$o,member(A,Xb),S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
             => ? [B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),Xb)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B3,Xb)),S) ) ) ) ) ) ).

% open_left
tff(fact_7105_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
           => ( ! [X2: A] :
                  ( aa(A,$o,P,X2)
                 => ( ! [Y5: A] :
                        ( aa(A,$o,P,Y5)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),X2) )
                   => aa(A,$o,Q,X2) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_7106_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Xb) )
           => ( order_Greatest(A,P) = Xb ) ) ) ) ).

% Greatest_equality
tff(fact_7107_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.head
tff(fact_7108_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.head
tff(fact_7109_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_7110_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_7111_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_7112_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or3652927894154168847AtMost(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_7113_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A)] :
          ( ! [A4: A,B3: A,X2: A] :
              ( aa(set(A),$o,member(A,A4),S)
             => ( aa(set(A),$o,member(A,B3),S)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B3)
                   => aa(set(A),$o,member(A,X2),S) ) ) ) )
         => ? [A4: A,B3: A] :
              ( ( S = bot_bot(set(A)) )
              | ( S = top_top(set(A)) )
              | ( S = aa(A,set(A),set_ord_lessThan(A),B3) )
              | ( S = aa(A,set(A),set_ord_atMost(A),B3) )
              | ( S = aa(A,set(A),set_ord_greaterThan(A),A4) )
              | ( S = aa(A,set(A),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_7114_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ! [F3: fun(nat,A)] :
                ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,F3,N8))
               => ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N8)),B2)
                 => ( order_antimono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aod(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_7115_decseq__const,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A] : order_antimono(nat,A,aTP_Lamp_aoe(A,fun(nat,A),K)) ) ).

% decseq_const
tff(fact_7116_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( aa(set(A),$o,member(A,I2),aa(A,set(A),set_ord_atLeast(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),I2) ) ) ).

% atLeast_iff
tff(fact_7117_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),Xb)),aa(A,set(A),set_ord_atLeast(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% atLeast_subset_iff
tff(fact_7118_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),aa(A,set(A),set_ord_atLeast(A),I2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),K),I2)) ) ).

% image_add_atLeast
tff(fact_7119_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or3652927894154168847AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).

% card_greaterThanAtMost_int
tff(fact_7120_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atLeast(A),L2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L2),L) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_7121_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),C2)),aa(A,set(A),set_ord_atLeast(A),A2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A2)) ) ).

% image_minus_const_atLeast
tff(fact_7122_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),minus_minus(A),C2)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ).

% image_minus_const_AtMost
tff(fact_7123_INT__decseq__offset,axiom,
    ! [A: $tType,F4: fun(nat,set(A)),Nb: nat] :
      ( order_antimono(nat,set(A),F4)
     => ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),F4),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),F4),aa(nat,set(nat),set_ord_atLeast(nat),Nb))) ) ) ).

% INT_decseq_offset
tff(fact_7124_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_greaterThan(nat),K) ).

% atLeast_Suc_greaterThan
tff(fact_7125_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_atLeast(A),L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_7126_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,Xb)) ) ) ) ).

% antimonoD
tff(fact_7127_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,Xb)) ) ) ) ).

% antimonoE
tff(fact_7128_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X2: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y4)),aa(A,B,F2,X2)) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_7129_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y3)),aa(A,B,F2,X)) ) ) ) ).

% antimono_def
tff(fact_7130_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I2: nat] :
          ( order_antimono(nat,A,A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,aa(nat,nat,suc,I2))),aa(nat,A,A3,I2)) ) ) ).

% decseq_SucD
tff(fact_7131_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N))),aa(nat,A,X5,N))
         => order_antimono(nat,A,X5) ) ) ).

% decseq_SucI
tff(fact_7132_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N4))),aa(nat,A,F2,N4)) ) ) ).

% decseq_Suc_iff
tff(fact_7133_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( order_antimono(nat,A,X5)
        <=> ! [M5: nat,N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N4)),aa(nat,A,X5,M5)) ) ) ) ).

% decseq_def
tff(fact_7134_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I2: nat,J: nat] :
          ( order_antimono(nat,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,J)),aa(nat,A,F2,I2)) ) ) ) ).

% decseqD
tff(fact_7135_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),A2)),aa(A,set(A),set_ord_greaterThan(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_7136_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_7137_decseq__bounded,axiom,
    ! [X5: fun(nat,real),B4: real] :
      ( order_antimono(nat,real,X5)
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B4),aa(nat,real,X5,I3))
       => bfun(nat,real,X5,at_top(nat)) ) ) ).

% decseq_bounded
tff(fact_7138_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),L5: A,Nb: nat] :
          ( order_antimono(nat,A,X5)
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L5),aa(nat,A,X5,Nb)) ) ) ) ).

% decseq_ge
tff(fact_7139_greaterThanAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or3652927894154168847AtMost(int,I2,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_7140_decseq__convergent,axiom,
    ! [X5: fun(nat,real),B4: real] :
      ( order_antimono(nat,real,X5)
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B4),aa(nat,real,X5,I3))
       => ~ ! [L6: real] :
              ( filterlim(nat,real,X5,topolo7230453075368039082e_nhds(real,L6),at_top(nat))
             => ~ ! [I: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),aa(nat,real,X5,I)) ) ) ) ).

% decseq_convergent
tff(fact_7141_UN__atLeast__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_atLeast(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atLeast_UNIV
tff(fact_7142_at__top__sub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_aof(A,filter(A))),aa(A,set(A),set_ord_atLeast(A),C2))) ) ).

% at_top_sub
tff(fact_7143_atLeast__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atLeast(nat),K)),aa(set(nat),set(nat),insert(nat,K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_7144_at__top__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_aog(A,filter(A))),top_top(set(A)))) ) ) ).

% at_top_def
tff(fact_7145_nhds__countable,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Xb: A] :
          ~ ! [X6: fun(nat,set(A))] :
              ( order_antimono(nat,set(A),X6)
             => ( ! [N8: nat] : aa(set(A),$o,topolo1002775350975398744n_open(A),aa(nat,set(A),X6,N8))
               => ( ! [N8: nat] : aa(set(A),$o,member(A,Xb),aa(nat,set(A),X6,N8))
                 => ( topolo7230453075368039082e_nhds(A,Xb) != aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(nat),set(filter(A)),image(nat,filter(A),aTP_Lamp_aoh(fun(nat,set(A)),fun(nat,filter(A)),X6)),top_top(set(nat)))) ) ) ) ) ) ).

% nhds_countable
tff(fact_7146_LIMSEQ__INF,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [X5: fun(nat,A)] :
          ( order_antimono(nat,A,X5)
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat))))),at_top(nat)) ) ) ).

% LIMSEQ_INF
tff(fact_7147_INF__Lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [X5: fun(nat,A),L: A] :
          ( order_antimono(nat,A,X5)
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L),at_top(nat))
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat)))) = L ) ) ) ) ).

% INF_Lim
tff(fact_7148_tendsto__at__right__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,X5: fun(A,B),L5: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ! [S5: fun(nat,A)] :
                ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,S5,N8))
               => ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S5,N8)),B2)
                 => ( order_antimono(nat,A,S5)
                   => ( filterlim(nat,A,S5,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aoi(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S5),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_7149_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y6: set(B),X5: set(A),F4: filter(B),F2: fun(A,C)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),Y6)),X5)
     => ( eventually(B,aTP_Lamp_aoj(set(B),fun(B,$o),Y6),F4)
       => ( map_filter_on(A,C,X5,F2,map_filter_on(B,A,Y6,G,F4)) = map_filter_on(B,C,Y6,aa(fun(B,A),fun(B,C),comp(A,C,B,F2),G),F4) ) ) ) ).

% map_filter_on_comp
tff(fact_7150_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F4: filter(A)] :
          ( topolo6773858410816713723filter(A,F4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [P5: fun(A,$o)] :
                  ( eventually(A,P5,F4)
                  & ! [X: A,Y3: A] :
                      ( ( aa(A,$o,P5,X)
                        & aa(A,$o,P5,Y3) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y3)),E4) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_7151_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X5: set(A),F4: filter(A),P: fun(B,$o),F2: fun(A,B)] :
      ( eventually(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),X5),F4)
     => ( eventually(B,P,map_filter_on(A,B,X5,F2,F4))
      <=> eventually(A,aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_aok(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),X5),P),F2),F4) ) ) ).

% eventually_map_filter_on
tff(fact_7152_nhds__generated__topology,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [T2: set(set(A)),Xb: A] :
          ( ( topolo1002775350975398744n_open(A) = topolo8378437560675496660pology(A,T2) )
         => ( topolo7230453075368039082e_nhds(A,Xb) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(set(A)),set(filter(A)),image(set(A),filter(A),principal(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(A,fun(set(A),$o),aTP_Lamp_aol(set(set(A)),fun(A,fun(set(A),$o)),T2),Xb)))) ) ) ) ).

% nhds_generated_topology
tff(fact_7153_continuous__at__Sup__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( order_antimono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Sup_Sup(A),S),aa(A,set(A),set_ord_lessThan(A),aa(set(A),A,complete_Sup_Sup(A),S))),F2)
           => ( ( S != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S)
               => ( aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),S)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),S)) ) ) ) ) ) ) ).

% continuous_at_Sup_antimono
tff(fact_7154_bdd__above_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A),M6: A] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),M6) )
         => condit941137186595557371_above(A,A3) ) ) ).

% bdd_above.I
tff(fact_7155_bdd__above__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I5: set(A),A3: fun(A,set(B))] :
          ( finite_finite2(A,I5)
         => ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)))
          <=> ! [X: A] :
                ( aa(set(A),$o,member(A,X),I5)
               => condit941137186595557371_above(B,aa(A,set(B),A3,X)) ) ) ) ) ).

% bdd_above_UN
tff(fact_7156_bdd__above_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A3: set(A),F2: fun(A,B),M6: B] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),M6) )
         => condit941137186595557371_above(B,aa(set(A),set(B),image(A,B,F2),A3)) ) ) ).

% bdd_above.I2
tff(fact_7157_bdd__above_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit941137186595557371_above(A,A3)
        <=> ? [M8: A] :
            ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),M8) ) ) ) ).

% bdd_above.unfold
tff(fact_7158_bdd__above_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit941137186595557371_above(A,A3)
         => ~ ! [M7: A] :
                ~ ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),M7) ) ) ) ).

% bdd_above.E
tff(fact_7159_cSup__upper2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A),Y: A] :
          ( aa(set(A),$o,member(A,Xb),X5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
           => ( condit941137186595557371_above(A,X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ) ).

% cSup_upper2
tff(fact_7160_cSup__upper,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A)] :
          ( aa(set(A),$o,member(A,Xb),X5)
         => ( condit941137186595557371_above(A,X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),X5)) ) ) ) ).

% cSup_upper
tff(fact_7161_cSUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Xb: A,A3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,member(A,Xb),A3)
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image(A,B,F2),A3))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ) ).

% cSUP_upper
tff(fact_7162_cSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Xb: B,U: A] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image(B,A,F2),A3))
         => ( aa(set(B),$o,member(B,Xb),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F2,Xb))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))) ) ) ) ) ).

% cSUP_upper2
tff(fact_7163_cSup__le__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A),A2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S)),A2)
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),S)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_7164_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( ( B4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A3)
           => ( ! [B3: A] :
                  ( aa(set(A),$o,member(A,B3),B4)
                 => ? [X4: A] :
                      ( aa(set(A),$o,member(A,X4),A3)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),X4) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B4)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ) ).

% cSup_mono
tff(fact_7165_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Y: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X5))
            <=> ? [X: A] :
                  ( aa(set(A),$o,member(A,X),X5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_7166_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I2: B] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image(B,A,F2),A3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),Y)
           => ( aa(set(B),$o,member(B,I2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,I2)),Y) ) ) ) ) ).

% cSUP_lessD
tff(fact_7167_generate__topology__Union,axiom,
    ! [B: $tType,A: $tType,I5: set(A),S: set(set(B)),K5: fun(A,set(B))] :
      ( ! [K2: A] :
          ( aa(set(A),$o,member(A,K2),I5)
         => aa(set(B),$o,topolo8378437560675496660pology(B,S),aa(A,set(B),K5,K2)) )
     => aa(set(B),$o,topolo8378437560675496660pology(B,S),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),K5),I5))) ) ).

% generate_topology_Union
tff(fact_7168_cSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),U: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image(A,B,F2),A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),U)
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),U) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_7169_cSUP__mono,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),G: fun(C,B),B4: set(C),F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(C),set(B),image(C,B,G),B4))
           => ( ! [N: A] :
                  ( aa(set(A),$o,member(A,N),A3)
                 => ? [X4: C] :
                      ( aa(set(C),$o,member(C,X4),B4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,N)),aa(C,B,G,X4)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image(C,B,G),B4))) ) ) ) ) ).

% cSUP_mono
tff(fact_7170_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,B4)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4)) ) ) ) ) ).

% cSup_subset_mono
tff(fact_7171_less__cSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A3: set(A),F2: fun(A,B),A2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image(A,B,F2),A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3)))
            <=> ? [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),aa(A,B,F2,X)) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_7172_cSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),G: fun(A,B),B4: set(A),F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image(A,B,G),B4))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G),B4))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_7173_cSUP__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A3: set(A),B4: fun(A,set(B)),F2: fun(B,C)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => ( aa(A,set(B),B4,X2) != bot_bot(set(B)) ) )
           => ( condit941137186595557371_above(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_aom(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B4),F2)),A3)))
             => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3)))) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aon(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B4),F2)),A3)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_7174_Bseq__bdd__above_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( bfun(nat,A,X5,at_top(nat))
         => condit941137186595557371_above(real,aa(set(nat),set(real),image(nat,real,aTP_Lamp_fq(fun(nat,A),fun(nat,real),X5)),top_top(set(nat)))) ) ) ).

% Bseq_bdd_above'
tff(fact_7175_continuous__at__Inf__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( order_antimono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Inf_Inf(A),S),aa(A,set(A),set_ord_greaterThan(A),aa(set(A),A,complete_Inf_Inf(A),S))),F2)
           => ( ( S != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S)
               => ( aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),S)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),S)) ) ) ) ) ) ) ).

% continuous_at_Inf_antimono
tff(fact_7176_MVT,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2)
               => differentiable(real,real,F2,topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
         => ? [L3: real,Z4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z4)
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z4),B2)
              & has_field_derivative(real,F2,L3,topolo174197925503356063within(real,Z4,top_top(set(real))))
              & ( aa(real,real,aa(real,fun(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,aa(real,fun(real,real),minus_minus(real),B2),A2)),L3) ) ) ) ) ) ).

% MVT
tff(fact_7177_bdd__belowI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A),Mb: A] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X2) )
         => condit1013018076250108175_below(A,A3) ) ) ).

% bdd_belowI
tff(fact_7178_bdd__below_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A),M6: A] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M6),X2) )
         => condit1013018076250108175_below(A,A3) ) ) ).

% bdd_below.I
tff(fact_7179_bdd__below__image__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aoo(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3))
        <=> ( condit1013018076250108175_below(A,aa(set(B),set(A),image(B,A,F2),A3))
            & condit1013018076250108175_below(A,aa(set(B),set(A),image(B,A,G),A3)) ) ) ) ).

% bdd_below_image_inf
tff(fact_7180_bdd__below__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I5: set(A),A3: fun(A,set(B))] :
          ( finite_finite2(A,I5)
         => ( condit1013018076250108175_below(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)))
          <=> ! [X: A] :
                ( aa(set(A),$o,member(A,X),I5)
               => condit1013018076250108175_below(B,aa(A,set(B),A3,X)) ) ) ) ) ).

% bdd_below_UN
tff(fact_7181_IVT_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),A2: B,Y: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,A2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,B2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A2,B2),F2)
               => ? [X2: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),B2)
                    & ( aa(B,A,F2,X2) = Y ) ) ) ) ) ) ) ).

% IVT'
tff(fact_7182_IVT2_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),B2: B,Y: A,A2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,B2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F2,A2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A2,B2),F2)
               => ? [X2: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X2)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),B2)
                    & ( aa(B,A,F2,X2) = Y ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_7183_cInf__lower2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A),Y: A] :
          ( aa(set(A),$o,member(A,Xb),X5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( condit1013018076250108175_below(A,X5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Y) ) ) ) ) ).

% cInf_lower2
tff(fact_7184_cInf__lower,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xb: A,X5: set(A)] :
          ( aa(set(A),$o,member(A,Xb),X5)
         => ( condit1013018076250108175_below(A,X5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Xb) ) ) ) ).

% cInf_lower
tff(fact_7185_bdd__below_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit1013018076250108175_below(A,A3)
         => ~ ! [M7: A] :
                ~ ! [X4: A] :
                    ( aa(set(A),$o,member(A,X4),A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M7),X4) ) ) ) ).

% bdd_below.E
tff(fact_7186_bdd__below_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: set(A)] :
          ( condit1013018076250108175_below(A,A3)
        <=> ? [M8: A] :
            ! [X: A] :
              ( aa(set(A),$o,member(A,X),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M8),X) ) ) ) ).

% bdd_below.unfold
tff(fact_7187_continuous__on__dist,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( topolo81223032696312382ous_on(A,B,S3,G)
           => topolo81223032696312382ous_on(A,real,S3,aa(fun(A,B),fun(A,real),aTP_Lamp_aop(fun(A,B),fun(fun(A,B),fun(A,real)),F2),G)) ) ) ) ).

% continuous_on_dist
tff(fact_7188_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_aoq(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G)) ) ) ) ).

% bounded_linear.continuous_on
tff(fact_7189_open__Collect__neq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topological_t2_space(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_aor(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% open_Collect_neq
tff(fact_7190_continuous__on__powr,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( topolo81223032696312382ous_on(A,real,S3,G)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S3)
                 => ( aa(A,real,F2,X2) != zero_zero(real) ) )
             => topolo81223032696312382ous_on(A,real,S3,aa(fun(A,real),fun(A,real),aTP_Lamp_aos(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr
tff(fact_7191_continuous__on__op__minus,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [S3: set(A),Xb: A] : topolo81223032696312382ous_on(A,A,S3,aa(A,fun(A,A),minus_minus(A),Xb)) ) ).

% continuous_on_op_minus
tff(fact_7192_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_7193_continuous__on__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_aot(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_minus
tff(fact_7194_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_aou(fun(real,real),fun(real,real),F2)) ) ).

% continuous_on_arsinh'
tff(fact_7195_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_aov(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_max
tff(fact_7196_continuous__on__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => topolo81223032696312382ous_on(A,B,A3,aTP_Lamp_aow(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_sinh
tff(fact_7197_continuous__on__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => topolo81223032696312382ous_on(A,B,A3,aTP_Lamp_aox(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_cosh
tff(fact_7198_continuous__on__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S3: set(A),F2: fun(A,real),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( topolo81223032696312382ous_on(A,B,S3,G)
           => topolo81223032696312382ous_on(A,B,S3,aa(fun(A,B),fun(A,B),aTP_Lamp_aoy(fun(A,real),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_scaleR
tff(fact_7199_continuous__on__real__root,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),Nb: nat] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => topolo81223032696312382ous_on(A,real,S3,aa(nat,fun(A,real),aTP_Lamp_aoz(fun(A,real),fun(nat,fun(A,real)),F2),Nb)) ) ) ).

% continuous_on_real_root
tff(fact_7200_continuous__on__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: set(A),Nb: nat] : topolo81223032696312382ous_on(A,A,A3,aTP_Lamp_abh(nat,fun(A,A),Nb)) ) ).

% continuous_on_pochhammer
tff(fact_7201_continuous__on__pochhammer_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),F2: fun(A,B),Nb: nat] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => topolo81223032696312382ous_on(A,B,S3,aa(nat,fun(A,B),aTP_Lamp_apa(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% continuous_on_pochhammer'
tff(fact_7202_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_aax(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_cos
tff(fact_7203_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_aaw(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_sin
tff(fact_7204_continuous__on__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_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_apb(fun(A,B),fun(A,B),F2)) ) ) ).

% continuous_on_exp
tff(fact_7205_continuous__on__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [S3: set(A),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,G)
         => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_apc(fun(A,real),fun(A,B),G)) ) ) ).

% continuous_on_of_real
tff(fact_7206_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_apd(B,fun(A,B),C2)) ) ).

% continuous_on_const
tff(fact_7207_continuous__on__id,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A)] : topolo81223032696312382ous_on(A,A,S3,aTP_Lamp_acf(A,A)) ) ).

% continuous_on_id
tff(fact_7208_continuous__on__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),F2: fun(A,B),Nb: nat] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => topolo81223032696312382ous_on(A,B,S3,aa(nat,fun(A,B),aTP_Lamp_ape(fun(A,B),fun(nat,fun(A,B)),F2),Nb)) ) ) ).

% continuous_on_power
tff(fact_7209_continuous__on__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,nat)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => ( topolo81223032696312382ous_on(A,nat,A3,G)
           => topolo81223032696312382ous_on(A,B,A3,aa(fun(A,nat),fun(A,B),aTP_Lamp_apf(fun(A,B),fun(fun(A,nat),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_power'
tff(fact_7210_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_apg(fun(A,B),fun(A,real),F2)) ) ) ).

% continuous_on_norm
tff(fact_7211_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_aph(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_real_sqrt
tff(fact_7212_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_api(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_arctan
tff(fact_7213_continuous__on__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),F2: fun(A,B),C2: B] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => topolo81223032696312382ous_on(A,B,S3,aa(B,fun(A,B),aTP_Lamp_apj(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_on_mult_right
tff(fact_7214_continuous__on__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),F2: fun(A,B),C2: B] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => topolo81223032696312382ous_on(A,B,S3,aa(B,fun(A,B),aTP_Lamp_apk(fun(A,B),fun(B,fun(A,B)),F2),C2)) ) ) ).

% continuous_on_mult_left
tff(fact_7215_continuous__on__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(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_apl(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_mult'
tff(fact_7216_continuous__on__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( topolo81223032696312382ous_on(A,B,S3,G)
           => topolo81223032696312382ous_on(A,B,S3,aa(fun(A,B),fun(A,B),aTP_Lamp_apm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_mult
tff(fact_7217_continuous__on__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( topolo81223032696312382ous_on(A,B,S3,G)
           => topolo81223032696312382ous_on(A,B,S3,aa(fun(A,B),fun(A,B),aTP_Lamp_apn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_add
tff(fact_7218_continuous__on__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( topolo81223032696312382ous_on(A,B,S3,G)
           => topolo81223032696312382ous_on(A,B,S3,aa(fun(A,B),fun(A,B),aTP_Lamp_apo(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% continuous_on_diff
tff(fact_7219_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_app(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G)) ) ) ) ).

% continuous_on_Pair
tff(fact_7220_continuous__on__fst,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [S3: set(A),F2: fun(A,product_prod(B,C))] :
          ( topolo81223032696312382ous_on(A,product_prod(B,C),S3,F2)
         => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_apq(fun(A,product_prod(B,C)),fun(A,B),F2)) ) ) ).

% continuous_on_fst
tff(fact_7221_continuous__on__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [S3: set(A),F2: fun(A,product_prod(B,C))] :
          ( topolo81223032696312382ous_on(A,product_prod(B,C),S3,F2)
         => topolo81223032696312382ous_on(A,C,S3,aTP_Lamp_apr(fun(A,product_prod(B,C)),fun(A,C),F2)) ) ) ).

% continuous_on_snd
tff(fact_7222_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( topolo81223032696312382ous_on(A,B,S3,G)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S3)
                 => ( aa(A,B,G,X2) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S3,aa(fun(A,B),fun(A,B),aTP_Lamp_aps(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_7223_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S3)
               => ( aa(A,B,F2,X2) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_apt(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_7224_continuous__on__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),S: set(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => topolo81223032696312382ous_on(B,C,S,aa(A,fun(B,C),F2,I3)) )
         => topolo81223032696312382ous_on(B,C,S,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_apv(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_on_sum
tff(fact_7225_continuous__on__ln,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S3)
               => ( aa(A,real,F2,X2) != zero_zero(real) ) )
           => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_apw(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_ln
tff(fact_7226_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_apx(fun(A,real),fun(A,real),F2)) ) ) ).

% continuous_on_rabs
tff(fact_7227_continuous__on__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S: set(A),S3: set(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),S)
             => topolo81223032696312382ous_on(B,C,S3,aa(A,fun(B,C),F2,I3)) )
         => topolo81223032696312382ous_on(B,C,S3,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_apz(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S),F2)) ) ) ).

% continuous_on_prod
tff(fact_7228_continuous__on__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),S: set(B),F2: fun(A,fun(B,C))] :
          ( ! [I3: A] :
              ( aa(set(A),$o,member(A,I3),I5)
             => topolo81223032696312382ous_on(B,C,S,aa(A,fun(B,C),F2,I3)) )
         => topolo81223032696312382ous_on(B,C,S,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aqb(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2)) ) ) ).

% continuous_on_prod'
tff(fact_7229_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S3)
               => ( aa(A,B,F2,X2) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_aqc(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_7230_continuous__on__compose2,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Ta: set(A),G: fun(A,B),S3: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(A,B,Ta,G)
         => ( topolo81223032696312382ous_on(C,A,S3,F2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F2),S3)),Ta)
             => topolo81223032696312382ous_on(C,B,S3,aa(fun(C,A),fun(C,B),aTP_Lamp_aqd(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2)) ) ) ) ) ).

% continuous_on_compose2
tff(fact_7231_bdd__below_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A3: set(A),M6: B,F2: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M6),aa(A,B,F2,X2)) )
         => condit1013018076250108175_below(B,aa(set(A),set(B),image(A,B,F2),A3)) ) ) ).

% bdd_below.I2
tff(fact_7232_bdd__belowI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A3: set(A),Mb: B,F2: fun(A,B)] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(A,B,F2,X2)) )
         => condit1013018076250108175_below(B,aa(set(A),set(B),image(A,B,F2),A3)) ) ) ).

% bdd_belowI2
tff(fact_7233_continuous__onI__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & dense_order(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(B,A),A3: set(B)] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),aa(set(B),set(A),image(B,A,F2),A3))
         => ( ! [X2: B,Y4: B] :
                ( aa(set(B),$o,member(B,X2),A3)
               => ( aa(set(B),$o,member(B,Y4),A3)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X2)),aa(B,A,F2,Y4)) ) ) )
           => topolo81223032696312382ous_on(B,A,A3,F2) ) ) ) ).

% continuous_onI_mono
tff(fact_7234_continuous__on__arcosh,axiom,
    ! [A3: set(real)] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),A3),aa(real,set(real),set_ord_atLeast(real),one_one(real)))
     => topolo81223032696312382ous_on(real,real,A3,arcosh(real)) ) ).

% continuous_on_arcosh
tff(fact_7235_cINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Xb: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image(B,A,F2),A3))
         => ( aa(set(B),$o,member(B,Xb),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(B,A,F2,Xb)) ) ) ) ).

% cINF_lower
tff(fact_7236_cINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Xb: B,U: A] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image(B,A,F2),A3))
         => ( aa(set(B),$o,member(B,Xb),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,Xb)),U)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),U) ) ) ) ) ).

% cINF_lower2
tff(fact_7237_le__cInf__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A),A2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,complete_Inf_Inf(A),S))
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),S)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_7238_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B4: set(A),A3: set(A)] :
          ( ( B4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A3)
           => ( ! [B3: A] :
                  ( aa(set(A),$o,member(A,B3),B4)
                 => ? [X4: A] :
                      ( aa(set(A),$o,member(A,X4),A3)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B3) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4)) ) ) ) ) ).

% cInf_mono
tff(fact_7239_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X5: set(A),Y: A] :
          ( ( X5 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X5)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X5)),Y)
            <=> ? [X: A] :
                  ( aa(set(A),$o,member(A,X),X5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_7240_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_aqe(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_7241_continuous__on__tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(C,A),L: A,F4: filter(C)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,L),F4)
           => ( aa(set(A),$o,member(A,L),S3)
             => ( eventually(C,aa(fun(C,A),fun(C,$o),aTP_Lamp_aqf(set(A),fun(fun(C,A),fun(C,$o)),S3),G),F4)
               => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aex(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),topolo7230453075368039082e_nhds(B,aa(A,B,F2,L)),F4) ) ) ) ) ) ).

% continuous_on_tendsto_compose
tff(fact_7242_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S3,F2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S3)
               => ( cos(A,aa(A,A,F2,X2)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S3,aTP_Lamp_abi(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_7243_open__Collect__less__Int,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( topolo81223032696312382ous_on(A,real,S3,G)
           => ? [A7: set(A)] :
                ( aa(set(A),$o,topolo1002775350975398744n_open(A),A7)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),S3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,real),fun(A,$o),aa(fun(A,real),fun(fun(A,real),fun(A,$o)),aTP_Lamp_aqg(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,$o))),S3),F2),G)) ) ) ) ) ) ).

% open_Collect_less_Int
tff(fact_7244_continuous__on__open__UN,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [S: set(A),A3: fun(A,set(B)),F2: fun(B,C)] :
          ( ! [S2: A] :
              ( aa(set(A),$o,member(A,S2),S)
             => aa(set(B),$o,topolo1002775350975398744n_open(B),aa(A,set(B),A3,S2)) )
         => ( ! [S2: A] :
                ( aa(set(A),$o,member(A,S2),S)
               => topolo81223032696312382ous_on(B,C,aa(A,set(B),A3,S2),F2) )
           => topolo81223032696312382ous_on(B,C,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),S)),F2) ) ) ) ).

% continuous_on_open_UN
tff(fact_7245_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S3,F2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S3)
               => ( sin(A,aa(A,A,F2,X2)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S3,aTP_Lamp_abj(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_7246_continuous__on__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A3,F2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => ( cosh(B,aa(A,B,F2,X2)) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,A3,aTP_Lamp_aqh(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_7247_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),Y: A,I2: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image(B,A,F2),A3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)))
           => ( aa(set(B),$o,member(B,I2),A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F2,I2)) ) ) ) ) ).

% less_cINF_D
tff(fact_7248_continuous__on__arcosh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => ( ! [X2: real] :
            ( aa(set(real),$o,member(real,X2),A3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,F2,X2)) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_aqi(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_arcosh'
tff(fact_7249_continuous__image__closed__interval,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ? [C3: real,D3: real] :
            ( ( aa(set(real),set(real),image(real,real,F2),set_or1337092689740270186AtMost(real,A2,B2)) = set_or1337092689740270186AtMost(real,C3,D3) )
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),D3) ) ) ) ).

% continuous_image_closed_interval
tff(fact_7250_le__cINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),U: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image(A,B,F2),A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3)))
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,X)) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_7251_cINF__mono,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [B4: set(A),F2: fun(C,B),A3: set(C),G: fun(A,B)] :
          ( ( B4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(C),set(B),image(C,B,F2),A3))
           => ( ! [M: A] :
                  ( aa(set(A),$o,member(A,M),B4)
                 => ? [X4: C] :
                      ( aa(set(C),$o,member(C,X4),A3)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F2,X4)),aa(A,B,G,M)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image(C,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,G),B4))) ) ) ) ) ).

% cINF_mono
tff(fact_7252_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,B4)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B4)),aa(set(A),A,complete_Inf_Inf(A),A3)) ) ) ) ) ).

% cInf_superset_mono
tff(fact_7253_open__Collect__positive,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ? [A7: set(A)] :
              ( aa(set(A),$o,topolo1002775350975398744n_open(A),A7)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),S3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,real),fun(A,$o),aTP_Lamp_aqj(set(A),fun(fun(A,real),fun(A,$o)),S3),F2)) ) ) ) ) ).

% open_Collect_positive
tff(fact_7254_continuous__on__powr_H,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( topolo81223032696312382ous_on(A,real,S3,G)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F2,X2))
                    & ( ( aa(A,real,F2,X2) = zero_zero(real) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X2)) ) ) )
             => topolo81223032696312382ous_on(A,real,S3,aa(fun(A,real),fun(A,real),aTP_Lamp_aos(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_7255_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( topolo81223032696312382ous_on(A,real,S3,G)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,X2)) )
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),S3)
                   => ( aa(A,real,F2,X2) != one_one(real) ) )
               => ( ! [X2: A] :
                      ( aa(set(A),$o,member(A,X2),S3)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X2)) )
                 => topolo81223032696312382ous_on(A,real,S3,aa(fun(A,real),fun(A,real),aTP_Lamp_aqk(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_7256_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_7257_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_7258_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X2))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X2)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_aql(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arccos
tff(fact_7259_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X2))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X2)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_aqm(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arcsin
tff(fact_7260_cINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A3: set(A),F2: fun(A,B),A2: B] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image(A,B,F2),A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))),A2)
            <=> ? [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),A2) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_7261_cINF__inf__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image(A,B,F2),A3))
           => ( condit1013018076250108175_below(B,aa(set(A),set(B),image(A,B,G),A3))
             => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,G),A3))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aqn(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),A3)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_7262_continuous__on__artanh,axiom,
    ! [A3: set(real)] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),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_7263_cSUP__eq__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A3: set(B),A2: B] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image(B,A,F2),A3))
           => ( condit1013018076250108175_below(A,aa(set(B),set(A),image(B,A,F2),A3))
             => ( aa(set(B),$o,member(B,A2),A3)
               => ( aa(B,A,F2,A2) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3)) ) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_7264_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,B2: A,F2: fun(A,A)] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B2)
               => ? [Y5: A] : has_field_derivative(A,F2,Y5,topolo174197925503356063within(A,X2,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_7265_continuous__on__artanh_H,axiom,
    ! [A3: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F2)
     => ( ! [X2: real] :
            ( aa(set(real),$o,member(real,X2),A3)
           => aa(set(real),$o,member(real,aa(real,real,F2,X2)),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_aqo(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_artanh'
tff(fact_7266_cINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),G: fun(A,B),B4: set(A),F2: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image(A,B,G),B4))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
             => ( ! [X2: A] :
                    ( aa(set(A),$o,member(A,X2),B4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,G,X2)),aa(A,B,F2,X2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,G),B4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_7267_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F6: fun(real,fun(real,real))] :
      ( aa(real,$o,aa(real,fun(real,$o),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)
         => ( ! [X2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2)
                 => has_derivative(real,real,F2,aa(real,fun(real,real),F6,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
           => ? [Z4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z4),B2)
                & ! [X4: real] : aa(real,real,aa(real,fun(real,real),F6,Z4),X4) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_7268_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( condit1013018076250108175_below(A,A3)
         => ( condit1013018076250108175_below(A,B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_7269_mvt,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),F6: fun(real,fun(real,real))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2)
               => has_derivative(real,real,F2,aa(real,fun(real,real),F6,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
         => ~ ! [Xi: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xi)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xi),B2)
                 => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) != aa(real,real,aa(real,fun(real,real),F6,Xi),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)) ) ) ) ) ) ) ).

% mvt
tff(fact_7270_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A3)
           => ( condit1013018076250108175_below(A,A3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3)) ) ) ) ) ).

% cInf_le_cSup
tff(fact_7271_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),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_aqp(A,B)) ) ).

% continuous_on_of_int_floor
tff(fact_7272_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),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),ring_1_Ints(A)),aTP_Lamp_aqq(A,B)) ) ).

% continuous_on_of_int_ceiling
tff(fact_7273_cINF__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A3: set(A),B4: fun(A,set(B)),F2: fun(B,C)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => ( aa(A,set(B),B4,X2) != bot_bot(set(B)) ) )
           => ( condit1013018076250108175_below(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_aom(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B4),F2)),A3)))
             => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image(B,C,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3)))) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aqr(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B4),F2)),A3)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_7274_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_7275_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_7276_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
         => ( aa(real,real,F2,B2) = aa(real,real,F2,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_7277_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y5),zero_zero(real)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,B2)),aa(real,real,F2,A2)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_7278_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2)
             => ? [Y5: real] :
                  ( has_field_derivative(real,F2,Y5,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y5) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,A2)),aa(real,real,F2,B2)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_7279_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F2: fun(real,real),Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F2)
       => ( ! [X2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xb)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),B2)
             => ( aa(real,real,F2,Xb) = aa(real,real,F2,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_7280_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A2: A,B2: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B2)),topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2)))
           => ( ! [X2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),B2)
                   => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X2)),topolo174197925503356063within(A,X2,top_top(set(A)))) ) )
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_7281_Rolle,axiom,
    ! [A2: real,B2: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),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)
         => ( ! [X2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X2),B2)
                 => differentiable(real,real,F2,topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
           => ? [Z4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z4),B2)
                & has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,Z4,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_7282_LIMSEQ__decseq__INF,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [X5: fun(nat,A)] :
          ( condit1013018076250108175_below(A,aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat))))
         => ( order_antimono(nat,A,X5)
           => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat))))),at_top(nat)) ) ) ) ).

% LIMSEQ_decseq_INF
tff(fact_7283_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),P: fun(A,$o)] : least(A,Less_eq,P) = the(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aqs(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Less_eq),P)) ).

% ord.Least_def
tff(fact_7284_eventually__filtercomap__at__topological,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,$o),F2: fun(A,B),A3: B,B4: set(B)] :
          ( eventually(A,P,filtercomap(A,B,F2,topolo174197925503356063within(B,A3,B4)))
        <=> ? [S9: set(B)] :
              ( aa(set(B),$o,topolo1002775350975398744n_open(B),S9)
              & aa(set(B),$o,member(B,A3),S9)
              & ! [X: A] :
                  ( aa(set(B),$o,member(B,aa(A,B,F2,X)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S9),B4)),aa(set(B),set(B),insert(B,A3),bot_bot(set(B)))))
                 => aa(A,$o,P,X) ) ) ) ) ).

% eventually_filtercomap_at_topological
tff(fact_7285_eventually__filtercomapI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F4: filter(A),F2: fun(B,A)] :
      ( eventually(A,P,F4)
     => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_akr(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2),filtercomap(B,A,F2,F4)) ) ).

% eventually_filtercomapI
tff(fact_7286_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_aqt(fun(A,real),fun(A,complex),G))
        <=> topolo81223032696312382ous_on(A,real,S,G) ) ) ).

% continuous_on_of_real_o_iff
tff(fact_7287_continuous__on__cnj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),G: fun(A,complex)] :
          ( topolo81223032696312382ous_on(A,complex,S3,G)
         => topolo81223032696312382ous_on(A,complex,S3,aTP_Lamp_aqu(fun(A,complex),fun(A,complex),G)) ) ) ).

% continuous_on_cnj
tff(fact_7288_continuous__on__Im,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),G: fun(A,complex)] :
          ( topolo81223032696312382ous_on(A,complex,S3,G)
         => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_aqv(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_on_Im
tff(fact_7289_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_aqw(fun(A,real),fun(A,complex),F2)) ) ) ).

% continuous_on_cis
tff(fact_7290_continuous__on__Re,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),G: fun(A,complex)] :
          ( topolo81223032696312382ous_on(A,complex,S3,G)
         => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_aqx(fun(A,complex),fun(A,real),G)) ) ) ).

% continuous_on_Re
tff(fact_7291_filtercomap__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filtercomap(A,A,aTP_Lamp_ml(A,A),F4) = F4 ).

% filtercomap_ident
tff(fact_7292_filtercomap__filtercomap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(A,B),G: fun(B,C),F4: filter(C)] : filtercomap(A,B,F2,filtercomap(B,C,G,F4)) = filtercomap(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aqy(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),F4) ).

% filtercomap_filtercomap
tff(fact_7293_filtercomap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(A,B),F4: fun(C,filter(B)),B4: set(C)] : filtercomap(A,B,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image(C,filter(B),F4),B4))) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aqz(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F4)),B4)) ).

% filtercomap_INF
tff(fact_7294_eventually__filtercomap__at__top__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_top(B)))
        <=> ? [N5: B] :
            ! [X: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N5),aa(A,B,F2,X))
             => aa(A,$o,P,X) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_7295_eventually__filtercomap__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_top(B)))
        <=> ? [N5: B] :
            ! [X: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),N5),aa(A,B,F2,X))
             => aa(A,$o,P,X) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_7296_eventually__filtercomap__at__bot__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_bot(B)))
        <=> ? [N5: B] :
            ! [X: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),N5)
             => aa(A,$o,P,X) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_7297_eventually__filtercomap__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F2,at_bot(B)))
        <=> ? [N5: B] :
            ! [X: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),N5)
             => aa(A,$o,P,X) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_7298_filtercomap__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(A,C),F4: fun(B,filter(C)),B4: set(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_ara(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F2),F4)),B4))),filtercomap(A,C,F2,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),image(B,filter(C),F4),B4)))) ).

% filtercomap_SUP
tff(fact_7299_Succ__def,axiom,
    ! [A: $tType,Kl: set(list(A)),Kl2: list(A)] : bNF_Greatest_Succ(A,Kl,Kl2) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_arb(set(list(A)),fun(list(A),fun(A,$o)),Kl),Kl2)) ).

% Succ_def
tff(fact_7300_uniformity__dist,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ( topolo7806501430040627800ormity(A) = aa(set(filter(product_prod(A,A))),filter(product_prod(A,A)),complete_Inf_Inf(filter(product_prod(A,A))),aa(set(real),set(filter(product_prod(A,A))),image(real,filter(product_prod(A,A)),aTP_Lamp_ard(real,filter(product_prod(A,A)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ).

% uniformity_dist
tff(fact_7301_uniformity__transE,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ~ ! [D8: fun(product_prod(A,A),$o)] :
                ( eventually(product_prod(A,A),D8,topolo7806501430040627800ormity(A))
               => ~ ! [X4: A,Y5: A] :
                      ( aa(product_prod(A,A),$o,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5))
                     => ! [Z2: A] :
                          ( aa(product_prod(A,A),$o,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z2))
                         => aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z2)) ) ) ) ) ) ).

% uniformity_transE
tff(fact_7302_uniformity__trans,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => ? [D8: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),D8,topolo7806501430040627800ormity(A))
              & ! [X4: A,Y5: A,Z2: A] :
                  ( aa(product_prod(A,A),$o,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5))
                 => ( aa(product_prod(A,A),$o,D8,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z2))
                   => aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z2)) ) ) ) ) ) ).

% uniformity_trans
tff(fact_7303_uniformity__refl,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o),Xb: A] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => aa(product_prod(A,A),$o,E5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Xb)) ) ) ).

% uniformity_refl
tff(fact_7304_uniformity__sym,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(A,A),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_are(fun(product_prod(A,A),$o),fun(A,fun(A,$o)),E5)),topolo7806501430040627800ormity(A)) ) ) ).

% uniformity_sym
tff(fact_7305_open__uniformity,axiom,
    ! [A: $tType] :
      ( topolo569519726778239578ormity(A)
     => ! [U3: set(A)] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),U3)
        <=> ! [X: A] :
              ( aa(set(A),$o,member(A,X),U3)
             => eventually(product_prod(A,A),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_arf(set(A),fun(A,fun(A,fun(A,$o))),U3),X)),topolo7806501430040627800ormity(A)) ) ) ) ).

% open_uniformity
tff(fact_7306_Cauchy__uniform__iff,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [P5: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),P5,topolo7806501430040627800ormity(A))
             => ? [N5: nat] :
                ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N4)
                 => ! [M5: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),M5)
                     => aa(product_prod(A,A),$o,P5,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,X5,N4)),aa(nat,A,X5,M5))) ) ) ) ) ) ).

% Cauchy_uniform_iff
tff(fact_7307_totally__bounded__def,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S: set(A)] :
          ( topolo6688025880775521714ounded(A,S)
        <=> ! [E6: fun(product_prod(A,A),$o)] :
              ( eventually(product_prod(A,A),E6,topolo7806501430040627800ormity(A))
             => ? [X7: set(A)] :
                  ( finite_finite2(A,X7)
                  & ! [X: A] :
                      ( aa(set(A),$o,member(A,X),S)
                     => ? [Xa2: A] :
                          ( aa(set(A),$o,member(A,Xa2),X7)
                          & aa(product_prod(A,A),$o,E6,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X)) ) ) ) ) ) ) ).

% totally_bounded_def
tff(fact_7308_eventually__nhds__uniformity,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,Xb))
        <=> eventually(product_prod(A,A),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_arg(fun(A,$o),fun(A,fun(A,fun(A,$o))),P),Xb)),topolo7806501430040627800ormity(A)) ) ) ).

% eventually_nhds_uniformity
tff(fact_7309_eventually__uniformity__metric,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [P: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),P,topolo7806501430040627800ormity(A))
        <=> ? [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
              & ! [X: A,Y3: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y3)),E4)
                 => aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)) ) ) ) ) ).

% eventually_uniformity_metric
tff(fact_7310_tendsto__iff__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [F2: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E6: fun(product_prod(B,B),$o)] :
              ( eventually(product_prod(B,B),E6,topolo7806501430040627800ormity(B))
             => eventually(A,aa(fun(product_prod(B,B),$o),fun(A,$o),aa(B,fun(fun(product_prod(B,B),$o),fun(A,$o)),aTP_Lamp_arh(fun(A,B),fun(B,fun(fun(product_prod(B,B),$o),fun(A,$o))),F2),L),E6),F4) ) ) ) ).

% tendsto_iff_uniformity
tff(fact_7311_uniformity__real__def,axiom,
    topolo7806501430040627800ormity(real) = aa(set(filter(product_prod(real,real))),filter(product_prod(real,real)),complete_Inf_Inf(filter(product_prod(real,real))),aa(set(real),set(filter(product_prod(real,real))),image(real,filter(product_prod(real,real)),aTP_Lamp_arj(real,filter(product_prod(real,real)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_real_def
tff(fact_7312_uniformity__complex__def,axiom,
    topolo7806501430040627800ormity(complex) = aa(set(filter(product_prod(complex,complex))),filter(product_prod(complex,complex)),complete_Inf_Inf(filter(product_prod(complex,complex))),aa(set(real),set(filter(product_prod(complex,complex))),image(real,filter(product_prod(complex,complex)),aTP_Lamp_arl(real,filter(product_prod(complex,complex)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_complex_def
tff(fact_7313_compactE__image,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),C5: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ! [T5: B] :
                ( aa(set(B),$o,member(B,T5),C5)
               => aa(set(A),$o,topolo1002775350975398744n_open(A),aa(B,set(A),F2,T5)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),C5)))
             => ~ ! [C9: set(B)] :
                    ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C9),C5)
                   => ( finite_finite2(B,C9)
                     => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),C9))) ) ) ) ) ) ) ).

% compactE_image
tff(fact_7314_prod__filter__INF2,axiom,
    ! [C: $tType,B: $tType,A: $tType,J4: set(A),A3: filter(B),B4: fun(A,filter(C))] :
      ( ( J4 != bot_bot(set(A)) )
     => ( prod_filter(B,C,A3,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image(A,filter(C),B4),J4))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_arm(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A3),B4)),J4)) ) ) ).

% prod_filter_INF2
tff(fact_7315_open__complex__def,axiom,
    ! [U3: set(complex)] :
      ( aa(set(complex),$o,topolo1002775350975398744n_open(complex),U3)
    <=> ! [X: complex] :
          ( aa(set(complex),$o,member(complex,X),U3)
         => eventually(product_prod(complex,complex),aa(fun(complex,fun(complex,$o)),fun(product_prod(complex,complex),$o),product_case_prod(complex,complex,$o),aa(complex,fun(complex,fun(complex,$o)),aTP_Lamp_arn(set(complex),fun(complex,fun(complex,fun(complex,$o))),U3),X)),topolo7806501430040627800ormity(complex)) ) ) ).

% open_complex_def
tff(fact_7316_open__real__def,axiom,
    ! [U3: set(real)] :
      ( aa(set(real),$o,topolo1002775350975398744n_open(real),U3)
    <=> ! [X: real] :
          ( aa(set(real),$o,member(real,X),U3)
         => eventually(product_prod(real,real),aa(fun(real,fun(real,$o)),fun(product_prod(real,real),$o),product_case_prod(real,real,$o),aa(real,fun(real,fun(real,$o)),aTP_Lamp_aro(set(real),fun(real,fun(real,fun(real,$o))),U3),X)),topolo7806501430040627800ormity(real)) ) ) ).

% open_real_def
tff(fact_7317_eventually__prod__same,axiom,
    ! [A: $tType,P: fun(product_prod(A,A),$o),F4: filter(A)] :
      ( eventually(product_prod(A,A),P,prod_filter(A,A,F4,F4))
    <=> ? [Q7: fun(A,$o)] :
          ( eventually(A,Q7,F4)
          & ! [X: A,Y3: A] :
              ( aa(A,$o,Q7,X)
             => ( aa(A,$o,Q7,Y3)
               => aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)) ) ) ) ) ).

% eventually_prod_same
tff(fact_7318_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,P: fun(product_prod(A,B),$o),F4: filter(A),G7: filter(B)] :
      ( eventually(product_prod(A,B),P,prod_filter(A,B,F4,G7))
    <=> ? [Pf: fun(A,$o),Pg: fun(B,$o)] :
          ( eventually(A,Pf,F4)
          & eventually(B,Pg,G7)
          & ! [X: A,Y3: B] :
              ( aa(A,$o,Pf,X)
             => ( aa(B,$o,Pg,Y3)
               => aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)) ) ) ) ) ).

% eventually_prod_filter
tff(fact_7319_compact__attains__inf,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
                & ! [Xa: A] :
                    ( aa(set(A),$o,member(A,Xa),S)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa) ) ) ) ) ) ).

% compact_attains_inf
tff(fact_7320_compact__attains__sup,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X2: A] :
                ( aa(set(A),$o,member(A,X2),S)
                & ! [Xa: A] :
                    ( aa(set(A),$o,member(A,Xa),S)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X2) ) ) ) ) ) ).

% compact_attains_sup
tff(fact_7321_compact__diff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T2: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( aa(set(A),$o,topolo1002775350975398744n_open(A),T2)
           => topolo2193935891317330818ompact(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) ) ) ) ).

% compact_diff
tff(fact_7322_eventually__prod__sequentially,axiom,
    ! [P: fun(product_prod(nat,nat),$o)] :
      ( eventually(product_prod(nat,nat),P,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N5: nat] :
        ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),M5)
         => ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N4)
             => aa(product_prod(nat,nat),$o,P,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N4),M5)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_7323_eventually__prodI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F4: filter(A),Q: fun(B,$o),G7: filter(B)] :
      ( eventually(A,P,F4)
     => ( eventually(B,Q,G7)
       => eventually(product_prod(A,B),aa(fun(B,$o),fun(product_prod(A,B),$o),aTP_Lamp_arp(fun(A,$o),fun(fun(B,$o),fun(product_prod(A,B),$o)),P),Q),prod_filter(A,B,F4,G7)) ) ) ).

% eventually_prodI
tff(fact_7324_eventually__prod1,axiom,
    ! [A: $tType,B: $tType,B4: filter(A),P: fun(B,$o),A3: filter(B)] :
      ( ( B4 != bot_bot(filter(A)) )
     => ( eventually(product_prod(B,A),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),aTP_Lamp_arq(fun(B,$o),fun(B,fun(A,$o)),P)),prod_filter(B,A,A3,B4))
      <=> eventually(B,P,A3) ) ) ).

% eventually_prod1
tff(fact_7325_eventually__prod2,axiom,
    ! [A: $tType,B: $tType,A3: filter(A),P: fun(B,$o),B4: filter(B)] :
      ( ( A3 != bot_bot(filter(A)) )
     => ( eventually(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_arr(fun(B,$o),fun(A,fun(B,$o)),P)),prod_filter(A,B,A3,B4))
      <=> eventually(B,P,B4) ) ) ).

% eventually_prod2
tff(fact_7326_continuous__attains__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ? [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S3)
                  & ! [Xa: A] :
                      ( aa(set(A),$o,member(A,Xa),S3)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Xa)) ) ) ) ) ) ) ).

% continuous_attains_inf
tff(fact_7327_continuous__attains__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ? [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S3)
                  & ! [Xa: A] :
                      ( aa(set(A),$o,member(A,Xa),S3)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,X2)) ) ) ) ) ) ) ).

% continuous_attains_sup
tff(fact_7328_nhds__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [A2: A,B2: B] : topolo7230453075368039082e_nhds(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) = prod_filter(A,B,topolo7230453075368039082e_nhds(A,A2),topolo7230453075368039082e_nhds(B,B2)) ) ).

% nhds_prod
tff(fact_7329_filterlim__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G7: filter(B),F4: filter(A),G: fun(A,C),H6: filter(C)] :
      ( filterlim(A,B,F2,G7,F4)
     => ( filterlim(A,C,G,H6,F4)
       => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ars(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),prod_filter(B,C,G7,H6),F4) ) ) ).

% filterlim_Pair
tff(fact_7330_tendsto__mult__Pair,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [A2: A,B2: A] : filterlim(product_prod(A,A),A,aTP_Lamp_art(product_prod(A,A),A),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),prod_filter(A,A,topolo7230453075368039082e_nhds(A,A2),topolo7230453075368039082e_nhds(A,B2))) ) ).

% tendsto_mult_Pair
tff(fact_7331_tendsto__add__Pair,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [A2: A,B2: A] : filterlim(product_prod(A,A),A,aTP_Lamp_aru(product_prod(A,A),A),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),prod_filter(A,A,topolo7230453075368039082e_nhds(A,A2),topolo7230453075368039082e_nhds(A,B2))) ) ).

% tendsto_add_Pair
tff(fact_7332_uniformity__trans_H,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [E5: fun(product_prod(A,A),$o)] :
          ( eventually(product_prod(A,A),E5,topolo7806501430040627800ormity(A))
         => eventually(product_prod(product_prod(A,A),product_prod(A,A)),aa(fun(product_prod(A,A),fun(product_prod(A,A),$o)),fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),product_case_prod(product_prod(A,A),product_prod(A,A),$o),aa(fun(A,fun(A,fun(product_prod(A,A),$o))),fun(product_prod(A,A),fun(product_prod(A,A),$o)),product_case_prod(A,A,fun(product_prod(A,A),$o)),aTP_Lamp_arw(fun(product_prod(A,A),$o),fun(A,fun(A,fun(product_prod(A,A),$o))),E5))),prod_filter(product_prod(A,A),product_prod(A,A),topolo7806501430040627800ormity(A),topolo7806501430040627800ormity(A))) ) ) ).

% uniformity_trans'
tff(fact_7333_prod__filter__def,axiom,
    ! [A: $tType,B: $tType,F4: filter(A),G7: filter(B)] : prod_filter(A,B,F4,G7) = aa(set(filter(product_prod(A,B))),filter(product_prod(A,B)),complete_Inf_Inf(filter(product_prod(A,B))),aa(set(product_prod(fun(A,$o),fun(B,$o))),set(filter(product_prod(A,B))),image(product_prod(fun(A,$o),fun(B,$o)),filter(product_prod(A,B)),aa(fun(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))),fun(product_prod(fun(A,$o),fun(B,$o)),filter(product_prod(A,B))),product_case_prod(fun(A,$o),fun(B,$o),filter(product_prod(A,B))),aTP_Lamp_ary(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))))),aa(fun(product_prod(fun(A,$o),fun(B,$o)),$o),set(product_prod(fun(A,$o),fun(B,$o))),collect(product_prod(fun(A,$o),fun(B,$o))),aa(fun(fun(A,$o),fun(fun(B,$o),$o)),fun(product_prod(fun(A,$o),fun(B,$o)),$o),product_case_prod(fun(A,$o),fun(B,$o),$o),aa(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o)),aTP_Lamp_arz(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),F4),G7))))) ).

% prod_filter_def
tff(fact_7334_prod__filter__INF,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,I5: set(A),J4: set(B),A3: fun(A,filter(C)),B4: fun(B,filter(D))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ( J4 != bot_bot(set(B)) )
       => ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image(A,filter(C),A3),I5)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image(B,filter(D),B4),J4))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),image(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_asb(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J4),A3),B4)),I5)) ) ) ) ).

% prod_filter_INF
tff(fact_7335_prod__filter__INF1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A3: fun(A,filter(B)),B4: filter(C)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( prod_filter(B,C,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),A3),I5)),B4) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_asc(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A3),B4)),I5)) ) ) ).

% prod_filter_INF1
tff(fact_7336_sequentially__imp__eventually__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,A2: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( ! [F3: fun(nat,A)] :
                ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,F3,N8))
               => ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N8)),A2)
                 => ( order_mono(nat,A,F3)
                   => ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aod(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F3),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_7337_compact__imp__fip__image,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),I5: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ! [I3: B] :
                ( aa(set(B),$o,member(B,I3),I5)
               => topolo7761053866217962861closed(A,aa(B,set(A),F2,I3)) )
           => ( ! [I6: set(B)] :
                  ( finite_finite2(B,I6)
                 => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),I6),I5)
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),I6))) != bot_bot(set(A)) ) ) )
             => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),F2),I5))) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip_image
tff(fact_7338_incseq__const,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A] : order_mono(nat,A,aTP_Lamp_aoe(A,fun(nat,A),K)) ) ).

% incseq_const
tff(fact_7339_open__Diff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T2: set(A)] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),S)
         => ( topolo7761053866217962861closed(A,T2)
           => aa(set(A),$o,topolo1002775350975398744n_open(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) ) ) ) ).

% open_Diff
tff(fact_7340_closed__Diff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T2: set(A)] :
          ( topolo7761053866217962861closed(A,S)
         => ( aa(set(A),$o,topolo1002775350975398744n_open(A),T2)
           => topolo7761053866217962861closed(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) ) ) ) ).

% closed_Diff
tff(fact_7341_closed__INT,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A3: set(A),B4: fun(A,set(B))] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),A3)
             => topolo7761053866217962861closed(B,aa(A,set(B),B4,X2)) )
         => topolo7761053866217962861closed(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ).

% closed_INT
tff(fact_7342_closed__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A3: set(A),B4: fun(A,set(B))] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => topolo7761053866217962861closed(B,aa(A,set(B),B4,X2)) )
           => topolo7761053866217962861closed(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B4),A3))) ) ) ) ).

% closed_UN
tff(fact_7343_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F2: fun(A,B),Mb: A,Nb: A,M4: B,N3: B] :
          ( order_mono(A,B,F2)
         => ( ( aa(set(A),set(B),image(A,B,F2),set_or7035219750837199246ssThan(A,Mb,Nb)) = set_or7035219750837199246ssThan(B,M4,N3) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb)
             => ( aa(A,B,F2,Mb) = M4 ) ) ) ) ) ).

% mono_image_least
tff(fact_7344_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),I2: nat,J: nat,Xb: A,Y: A] :
          ( order_mono(A,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,F2,Xb))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I2),F2),Xb)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J),F2),Y)) ) ) ) ) ) ).

% funpow_mono2
tff(fact_7345_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F2: fun(A,A),P2: A,K: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,P2)),P2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A))),P2) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_7346_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F2: fun(A,A),P2: A,K: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,F2,P2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_7347_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => order_mono(A,A,aa(A,fun(A,A),times_times(A),A2)) ) ) ).

% mono_mult
tff(fact_7348_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F2: fun(A,B),A3: A,B4: A] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B4))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A3)),aa(A,B,F2,B4))) ) ) ).

% mono_inf
tff(fact_7349_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mono_invE
tff(fact_7350_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% monoD
tff(fact_7351_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% monoE
tff(fact_7352_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X2: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) )
         => order_mono(A,B,F2) ) ) ).

% monoI
tff(fact_7353_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_mono(A,B,F2)
        <=> ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) ) ) ) ).

% mono_def
tff(fact_7354_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I2: nat] :
          ( order_mono(nat,A,A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,I2)),aa(nat,A,A3,aa(nat,nat,suc,I2))) ) ) ).

% incseq_SucD
tff(fact_7355_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N)),aa(nat,A,X5,aa(nat,nat,suc,N)))
         => order_mono(nat,A,X5) ) ) ).

% incseq_SucI
tff(fact_7356_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_mono(nat,A,F2)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4))) ) ) ).

% incseq_Suc_iff
tff(fact_7357_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( order_mono(nat,A,X5)
        <=> ! [M5: nat,N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M5)),aa(nat,A,X5,N4)) ) ) ) ).

% incseq_def
tff(fact_7358_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I2: nat,J: nat] :
          ( order_mono(nat,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,I2)),aa(nat,A,F2,J)) ) ) ) ).

% incseqD
tff(fact_7359_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),A3: A,B4: A,Nb: nat] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),A3)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),B4)) ) ) ) ).

% funpow_mono
tff(fact_7360_cclfp__lowerbound,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(A,A),A3: A] :
          ( order_mono(A,A,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,A3)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),order_532582986084564980_cclfp(A,F2)),A3) ) ) ) ).

% cclfp_lowerbound
tff(fact_7361_open__Collect__imp,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),P))
         => ( aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),Q))
           => aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_asd(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% open_Collect_imp
tff(fact_7362_open__Collect__neg,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o)] :
          ( topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),P))
         => aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aer(fun(A,$o),fun(A,$o),P))) ) ) ).

% open_Collect_neg
tff(fact_7363_closed__Collect__imp,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),P))
         => ( topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),Q))
           => topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_asd(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% closed_Collect_imp
tff(fact_7364_closed__Collect__neg,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o)] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),aa(fun(A,$o),set(A),collect(A),P))
         => topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aer(fun(A,$o),fun(A,$o),P))) ) ) ).

% closed_Collect_neg
tff(fact_7365_mono__times__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => order_mono(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb)) ) ).

% mono_times_nat
tff(fact_7366_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Nb: nat] :
          ( order_mono(A,A,F2)
         => order_mono(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2)) ) ) ).

% mono_pow
tff(fact_7367_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A] : order_mono(A,A,aa(A,fun(A,A),plus_plus(A),A2)) ) ).

% mono_add
tff(fact_7368_mono__Suc,axiom,
    order_mono(nat,nat,suc) ).

% mono_Suc
tff(fact_7369_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% mono_strict_invE
tff(fact_7370_closed__Collect__const,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: $o] : topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ahx($o,fun(A,$o),(P)))) ) ).

% closed_Collect_const
tff(fact_7371_closed__Collect__disj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),P))
         => ( topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),Q))
           => topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahw(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% closed_Collect_disj
tff(fact_7372_closed__Collect__conj,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),P))
         => ( topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),Q))
           => topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ) ) ).

% closed_Collect_conj
tff(fact_7373_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_mono(nat,A,aTP_Lamp_ase(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_7374_closed__Collect__all,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,fun(B,$o))] :
          ( ! [I3: A] : topolo7761053866217962861closed(B,aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),P,I3)))
         => topolo7761053866217962861closed(B,aa(fun(B,$o),set(B),collect(B),aTP_Lamp_asf(fun(A,fun(B,$o)),fun(B,$o),P))) ) ) ).

% closed_Collect_all
tff(fact_7375_decseq__eq__incseq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X5: fun(nat,A)] :
          ( order_antimono(nat,A,X5)
        <=> order_mono(nat,A,aTP_Lamp_gz(fun(nat,A),fun(nat,A),X5)) ) ) ).

% decseq_eq_incseq
tff(fact_7376_incseq__bounded,axiom,
    ! [X5: fun(nat,real),B4: real] :
      ( order_mono(nat,real,X5)
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,X5,I3)),B4)
       => bfun(nat,real,X5,at_top(nat)) ) ) ).

% incseq_bounded
tff(fact_7377_closed__Collect__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topological_t2_space(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_asg(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% closed_Collect_eq
tff(fact_7378_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ash(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,A3),I5)))) ) ) ).

% mono_SUP
tff(fact_7379_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ).

% mono_Sup
tff(fact_7380_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ).

% mono_Inf
tff(fact_7381_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,A3),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ash(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))) ) ) ).

% mono_INF
tff(fact_7382_closed__Collect__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_asi(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% closed_Collect_le
tff(fact_7383_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_antimono(nat,A,aTP_Lamp_asj(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_7384_incseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),L5: A,Nb: nat] :
          ( order_mono(nat,A,X5)
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,Nb)),L5) ) ) ) ).

% incseq_le
tff(fact_7385_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Mb: nat,Nb: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( order_mono(A,A,F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2),top_top(A))) ) ) ) ).

% funpow_increasing
tff(fact_7386_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Mb: nat,Nb: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( order_mono(A,A,F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Mb),F2),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),bot_bot(A))) ) ) ) ).

% funpow_decreasing
tff(fact_7387_incseq__convergent,axiom,
    ! [X5: fun(nat,real),B4: real] :
      ( order_mono(nat,real,X5)
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,X5,I3)),B4)
       => ~ ! [L6: real] :
              ( filterlim(nat,real,X5,topolo7230453075368039082e_nhds(real,L6),at_top(nat))
             => ~ ! [I: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,X5,I)),L6) ) ) ) ).

% incseq_convergent
tff(fact_7388_nhds__closed,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [Xb: A,A3: set(A)] :
          ( aa(set(A),$o,member(A,Xb),A3)
         => ( aa(set(A),$o,topolo1002775350975398744n_open(A),A3)
           => ? [A10: set(A)] :
                ( aa(set(A),$o,member(A,Xb),A10)
                & topolo7761053866217962861closed(A,A10)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A10),A3)
                & eventually(A,aTP_Lamp_ask(set(A),fun(A,$o),A10),topolo7230453075368039082e_nhds(A,Xb)) ) ) ) ) ).

% nhds_closed
tff(fact_7389_continuous__on__closed__Union,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [I5: set(A),U3: fun(A,set(B)),F2: fun(B,C)] :
          ( finite_finite2(A,I5)
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),I5)
               => topolo7761053866217962861closed(B,aa(A,set(B),U3,I3)) )
           => ( ! [I3: A] :
                  ( aa(set(A),$o,member(A,I3),I5)
                 => topolo81223032696312382ous_on(B,C,aa(A,set(B),U3,I3),F2) )
             => topolo81223032696312382ous_on(B,C,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),U3),I5)),F2) ) ) ) ) ).

% continuous_on_closed_Union
tff(fact_7390_Lim__in__closed__set,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F2: fun(B,A),F4: filter(B),L: A] :
          ( topolo7761053866217962861closed(A,S)
         => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_asl(set(A),fun(fun(B,A),fun(B,$o)),S),F2),F4)
           => ( ( F4 != bot_bot(filter(B)) )
             => ( filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F4)
               => aa(set(A),$o,member(A,L),S) ) ) ) ) ) ).

% Lim_in_closed_set
tff(fact_7391_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F2)
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image(C,A,A3),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_asm(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,A3),I5)))) ) ) ) ) ).

% mono_cSUP
tff(fact_7392_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => ( condit941137186595557371_above(A,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ) ).

% mono_cSup
tff(fact_7393_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F2)
         => ( condit1013018076250108175_below(A,A3)
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),A3))) ) ) ) ) ).

% mono_cInf
tff(fact_7394_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F2: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F2)
         => ( condit1013018076250108175_below(A,aa(set(C),set(A),image(C,A,A3),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,A3),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_asm(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A3)),I5))) ) ) ) ) ).

% mono_cINF
tff(fact_7395_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)
     => order_mono(nat,nat,aTP_Lamp_asn(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_7396_SUP__Lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [X5: fun(nat,A),L: A] :
          ( order_mono(nat,A,X5)
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L),at_top(nat))
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat)))) = L ) ) ) ) ).

% SUP_Lim
tff(fact_7397_LIMSEQ__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [X5: fun(nat,A)] :
          ( order_mono(nat,A,X5)
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat))))),at_top(nat)) ) ) ).

% LIMSEQ_SUP
tff(fact_7398_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( finite_finite2(A,aa(set(nat),set(A),image(nat,A,F2),top_top(set(nat))))
         => ( order_mono(nat,A,F2)
           => ( ! [N: nat] :
                  ( ( aa(nat,A,F2,N) = aa(nat,A,F2,aa(nat,nat,suc,N)) )
                 => ( aa(nat,A,F2,aa(nat,nat,suc,N)) = aa(nat,A,F2,aa(nat,nat,suc,aa(nat,nat,suc,N))) ) )
             => ? [N7: nat] :
                  ( ! [N8: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N7)
                     => ! [M2: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N7)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N8)
                           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,M2)),aa(nat,A,F2,N8)) ) ) )
                  & ! [N8: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N8)
                     => ( aa(nat,A,F2,N7) = aa(nat,A,F2,N8) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_7399_tendsto__at__left__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [B2: A,A2: A,X5: fun(A,B),L5: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( ! [S5: fun(nat,A)] :
                ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S5,N8)),A2)
               => ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,S5,N8))
                 => ( order_mono(nat,A,S5)
                   => ( filterlim(nat,A,S5,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aoi(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S5),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ).

% tendsto_at_left_sequentially
tff(fact_7400_LIMSEQ__incseq__SUP,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [X5: fun(nat,A)] :
          ( condit941137186595557371_above(A,aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat))))
         => ( order_mono(nat,A,X5)
           => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,X5),top_top(set(nat))))),at_top(nat)) ) ) ) ).

% LIMSEQ_incseq_SUP
tff(fact_7401_continuous__at__Sup__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( order_mono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Sup_Sup(A),S),aa(A,set(A),set_ord_lessThan(A),aa(set(A),A,complete_Sup_Sup(A),S))),F2)
           => ( ( S != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S)
               => ( aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),S)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),S)) ) ) ) ) ) ) ).

% continuous_at_Sup_mono
tff(fact_7402_continuous__at__Inf__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( order_mono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Inf_Inf(A),S),aa(A,set(A),set_ord_greaterThan(A),aa(set(A),A,complete_Inf_Inf(A),S))),F2)
           => ( ( S != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S)
               => ( aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),S)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F2),S)) ) ) ) ) ) ) ).

% continuous_at_Inf_mono
tff(fact_7403_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F5: fun(nat,nat)] :
          ( order_mono(nat,nat,F5)
          & ( aa(set(nat),set(nat),image(nat,nat,F5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
             => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),aa(nat,nat,F5,I4)) ) )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) )
              <=> ( aa(nat,nat,F5,I4) = aa(nat,nat,F5,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_7404_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),R2: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R2) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_asp(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),P),R2)))) ).

% same_fst_def
tff(fact_7405_remdups__adj__set,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),remdups_adj(A,Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% remdups_adj_set
tff(fact_7406_same__fstI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Xb: A,Y7: B,Y: B,R2: fun(A,set(product_prod(B,B)))] :
      ( aa(A,$o,P,Xb)
     => ( aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y7),Y)),aa(A,set(product_prod(B,B)),R2,Xb))
       => aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y7)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y))),same_fst(A,B,P,R2)) ) ) ).

% same_fstI
tff(fact_7407_remdups__adj__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% remdups_adj_length
tff(fact_7408_remdups__adj__adjacent,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I2) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I2)) ) ) ).

% remdups_adj_adjacent
tff(fact_7409_remdups__adj__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ( remdups_adj(A,Xs) = cons(A,Xb,nil(A)) )
     => ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xb) ) ) ).

% remdups_adj_singleton
tff(fact_7410_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))) ) ).

% remdups_adj_length_ge1
tff(fact_7411_tendsto__at__topI__sequentially__real,axiom,
    ! [F2: fun(real,real),Y: real] :
      ( order_mono(real,real,F2)
     => ( filterlim(nat,real,aTP_Lamp_asq(fun(real,real),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,Y),at_top(nat))
       => filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Y),at_top(real)) ) ) ).

% tendsto_at_topI_sequentially_real
tff(fact_7412_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X2))
     => ( order_mono(nat,real,F2)
       => ( order_strict_mono(nat,nat,G)
         => ( bfun(nat,real,aa(fun(nat,nat),fun(nat,real),aTP_Lamp_asr(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),F2),G),at_top(nat))
          <=> bfun(nat,real,F2,at_top(nat)) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_7413_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => inj_on(real,real,aTP_Lamp_uw(nat,fun(real,real),Nb),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_7414_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_7415_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_ass(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_7416_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A2: A,A3: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),insert(A,A2),A3))
    <=> ( inj_on(A,B,F2,A3)
        & ~ aa(set(B),$o,member(B,aa(A,B,F2,A2)),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_7417_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F2: fun(A,B)] :
          ( ! [X2: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4)
             => ( aa(set(A),$o,member(A,X2),A3)
               => ( aa(set(A),$o,member(A,Y4),A3)
                 => ( aa(A,B,F2,X2) != aa(A,B,F2,Y4) ) ) ) )
         => ( ! [X2: A,Y4: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => ( aa(set(A),$o,member(A,Y4),A3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y4)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X2) ) ) )
           => inj_on(A,B,F2,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_7418_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% strict_mono_less_eq
tff(fact_7419_strict__mono__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [R: fun(A,B),Mb: A,Nb: A] :
          ( order_strict_mono(A,B,R)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),Nb)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,R,Mb)),aa(A,B,R,Nb)) ) ) ) ).

% strict_mono_leD
tff(fact_7420_strict__mono__imp__increasing,axiom,
    ! [F2: fun(nat,nat),Nb: nat] :
      ( order_strict_mono(nat,nat,F2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,F2,Nb)) ) ).

% strict_mono_imp_increasing
tff(fact_7421_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_strict_mono(nat,A,F2)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4))) ) ) ).

% strict_mono_Suc_iff
tff(fact_7422_strict__mono__add,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A] : order_strict_mono(A,A,aTP_Lamp_mm(A,fun(A,A),K)) ) ).

% strict_mono_add
tff(fact_7423_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_7424_inj__on__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,A3)
     => inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) ) ).

% inj_on_diff
tff(fact_7425_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)) ) ) ) ).

% strict_monoD
tff(fact_7426_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X2: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,Y4)) )
         => order_strict_mono(A,B,F2) ) ) ).

% strict_monoI
tff(fact_7427_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
        <=> ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y3)) ) ) ) ).

% strict_mono_def
tff(fact_7428_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xb: A,Y: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% strict_mono_less
tff(fact_7429_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_7430_seq__monosub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [S3: fun(nat,A)] :
        ? [F3: fun(nat,nat)] :
          ( order_strict_mono(nat,nat,F3)
          & topological_monoseq(A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_ast(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),S3),F3)) ) ) ).

% seq_monosub
tff(fact_7431_strict__mono__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( order(C)
        & order(A)
        & order(B) )
     => ! [R: fun(A,B),S3: fun(C,A)] :
          ( order_strict_mono(A,B,R)
         => ( order_strict_mono(C,A,S3)
           => order_strict_mono(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_asu(fun(A,B),fun(fun(C,A),fun(C,B)),R),S3)) ) ) ) ).

% strict_mono_compose
tff(fact_7432_inj__on__id2,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,A,aTP_Lamp_ml(A,A),A3) ).

% inj_on_id2
tff(fact_7433_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aTP_Lamp_asv(A,fun(A,A),A2),A3) ) ).

% inj_on_add'
tff(fact_7434_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A),D5: set(B)] :
      ( finite_finite2(A,A3)
     => ( inj_on(B,A,F2,D5)
       => finite_finite2(B,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_asw(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A3),F2),D5))) ) ) ).

% finite_inverse_image_gen
tff(fact_7435_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(B,A)] :
      ( finite_finite2(A,A3)
     => ( inj_on(B,A,F2,top_top(set(B)))
       => finite_finite2(B,aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_asx(set(A),fun(fun(B,A),fun(B,$o)),A3),F2))) ) ) ).

% finite_inverse_image
tff(fact_7436_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : inj_on(A,A,aTP_Lamp_nr(A,fun(A,A),A2),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_7437_inj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Nb: nat] :
      ( inj_on(A,A,F2,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),top_top(set(A))) ) ).

% inj_fn
tff(fact_7438_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_7439_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X2: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Y4)
             => ( aa(A,B,F2,X2) != aa(A,B,F2,Y4) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_7440_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_asy(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_7441_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_asz(fun(A,B),fun(A,fun(C,B)),F2),top_top(set(A))) ) ).

% inj_fun
tff(fact_7442_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: A,Xb: A,B2: A,F2: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F2)
             => ( inj_on(A,B,F2,set_or1337092689740270186AtMost(A,A2,B2))
               => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A2)),aa(A,B,F2,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,B2)) )
                  | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B2)),aa(A,B,F2,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xb)),aa(A,B,F2,A2)) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_7443_eventually__subseq,axiom,
    ! [R: fun(nat,nat),P: fun(nat,$o)] :
      ( order_strict_mono(nat,nat,R)
     => ( eventually(nat,P,at_top(nat))
       => eventually(nat,aa(fun(nat,$o),fun(nat,$o),aTP_Lamp_ata(fun(nat,nat),fun(fun(nat,$o),fun(nat,$o)),R),P),at_top(nat)) ) ) ).

% eventually_subseq
tff(fact_7444_image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(B),set(B),aa(set(B),fun(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)) ) ) ).

% image_set_diff
tff(fact_7445_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(B),nat,finite_card(B),A3))
     => ~ inj_on(B,A,F2,A3) ) ).

% pigeonhole
tff(fact_7446_sum_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(A,B),A3: set(A)] :
          ( inj_on(A,B,G,A3)
         => ( aa(set(B),B,groups7311177749621191930dd_sum(B,B,aTP_Lamp_atb(B,B)),aa(set(A),set(B),image(A,B,G),A3)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) ) ) ) ).

% sum.image_eq
tff(fact_7447_prod_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(A,B),A3: set(A)] :
          ( inj_on(A,B,G,A3)
         => ( aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7121269368397514597t_prod(B,B),aTP_Lamp_atc(B,B)),aa(set(A),set(B),image(A,B,G),A3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3) ) ) ) ).

% prod.image_eq
tff(fact_7448_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C5: set(A),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),C5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),C5)
         => ( aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(B),set(B),aa(set(B),fun(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_7449_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_7450_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F2: fun(B,C)] :
      ( ! [I3: A,J2: A] :
          ( aa(set(A),$o,member(A,I3),I5)
         => ( aa(set(A),$o,member(A,J2),I5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,I3)),aa(A,set(B),A3,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I3)) ) ) )
     => ( ! [I3: A] :
            ( aa(set(A),$o,member(A,I3),I5)
           => inj_on(B,C,F2,aa(A,set(B),A3,I3)) )
       => inj_on(B,C,F2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_7451_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),F2: fun(B,C),A3: fun(A,set(B))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I3: A] :
            ( aa(set(A),$o,member(A,I3),I5)
           => inj_on(B,C,F2,aa(A,set(B),A3,I3)) )
       => inj_on(B,C,F2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) ) ) ).

% inj_on_INTER
tff(fact_7452_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),C5: set(A),A3: set(C),B4: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F2,C5)
     => ( ! [X2: C] :
            ( aa(set(C),$o,member(C,X2),A3)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(C,set(A),B4,X2)),C5) )
       => ( aa(set(C),$o,member(C,J),A3)
         => ( aa(set(A),set(B),image(A,B,F2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),B4),A3))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_ul(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F2),B4)),A3)) ) ) ) ) ).

% image_INT
tff(fact_7453_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(B,B4)
       => ( ? [F5: fun(A,B)] :
              ( inj_on(A,B,F5,A3)
              & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F5),A3)),B4) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_7454_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B4: set(B)] :
      ( inj_on(A,B,F2,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B4)
       => ( finite_finite2(B,B4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ).

% card_inj_on_le
tff(fact_7455_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(B,B4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4))
         => ? [F3: fun(A,B)] :
              ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),B4)
              & inj_on(A,B,F3,A3) ) ) ) ) ).

% card_le_inj
tff(fact_7456_log__inj,axiom,
    ! [B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => inj_on(real,real,log(B2),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))) ) ).

% log_inj
tff(fact_7457_summable__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N: nat] :
                ( ~ aa(set(nat),$o,member(nat,N),aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_atd(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2))
            <=> summable(A,F2) ) ) ) ) ).

% summable_mono_reindex
tff(fact_7458_sums__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A),C2: A] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N: nat] :
                ( ~ aa(set(nat),$o,member(nat,N),aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_atd(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2)),C2)
            <=> aa(A,$o,sums(A,F2),C2) ) ) ) ) ).

% sums_mono_reindex
tff(fact_7459_suminf__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A)] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N: nat] :
                ( ~ aa(set(nat),$o,member(nat,N),aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,A,F2,N) = zero_zero(A) ) )
           => ( suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ate(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2)) = suminf(A,F2) ) ) ) ) ).

% suminf_mono_reindex
tff(fact_7460_increasing__Bseq__subseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,nat)] :
          ( ! [X2: nat,Y4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Y4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,X2))),real_V7770717601297561774m_norm(A,aa(nat,A,F2,Y4))) )
         => ( order_strict_mono(nat,nat,G)
           => ( bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_anq(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F2),G),at_top(nat))
            <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ) ).

% increasing_Bseq_subseq_iff
tff(fact_7461_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,C: $tType,F4: filter(A),G7: filter(B),H6: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F4,G7),H6) = filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),product_case_prod(A,product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_atg(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)))),prod_filter(A,product_prod(B,C),F4,prod_filter(B,C,G7,H6))) ).

% prod_filter_assoc
tff(fact_7462_uniformly__continuous__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S3: set(A),F2: fun(A,B),E5: fun(product_prod(B,B),$o)] :
          ( topolo6026614971017936543ous_on(A,B,S3,F2)
         => ( eventually(product_prod(B,B),E5,topolo7806501430040627800ormity(B))
           => eventually(product_prod(A,A),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(product_prod(B,B),$o),fun(A,fun(A,$o)),aa(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o))),aTP_Lamp_ath(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o)))),S3),F2),E5)),topolo7806501430040627800ormity(A)) ) ) ) ).

% uniformly_continuous_onD
tff(fact_7463_filtermap__id_H,axiom,
    ! [A: $tType,X4: filter(A)] : filtermap(A,A,aTP_Lamp_ml(A,A),X4) = X4 ).

% filtermap_id'
tff(fact_7464_inj__on__diff__nat,axiom,
    ! [N2: set(nat),K: nat] :
      ( ! [N: nat] :
          ( aa(set(nat),$o,member(nat,N),N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) )
     => inj_on(nat,nat,aTP_Lamp_ol(nat,fun(nat,nat),K),N2) ) ).

% inj_on_diff_nat
tff(fact_7465_inj__Some,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,option(A),some(A),A3) ).

% inj_Some
tff(fact_7466_inj__Suc,axiom,
    ! [N2: set(nat)] : inj_on(nat,nat,suc,N2) ).

% inj_Suc
tff(fact_7467_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),X5: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_ati(fun(A,B),fun(A,product_prod(A,B)),F2),X5) ).

% inj_on_convol_ident
tff(fact_7468_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_7469_inj__singleton,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,set(A),aTP_Lamp_sb(A,set(A)),A3) ).

% inj_singleton
tff(fact_7470_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),F4: filter(C)] : aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),filtermap(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_atj(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G),F4)),prod_filter(A,B,filtermap(C,A,F2,F4),filtermap(C,B,G,F4))) ).

% filtermap_Pair
tff(fact_7471_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_atl(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_7472_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A3: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_ve(A,fun(B,product_prod(B,A)))),A3) ).

% swap_inj_on
tff(fact_7473_inj__split__Cons,axiom,
    ! [A: $tType,X5: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_atm(list(A),fun(A,list(A)))),X5) ).

% inj_split_Cons
tff(fact_7474_uniformly__continuous__on__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo7287701948861334536_space(B)
        & topolo7287701948861334536_space(C)
        & topolo7287701948861334536_space(A) )
     => ! [S3: set(A),G: fun(A,B),F2: fun(B,C)] :
          ( topolo6026614971017936543ous_on(A,B,S3,G)
         => ( topolo6026614971017936543ous_on(B,C,aa(set(A),set(B),image(A,B,G),S3),F2)
           => topolo6026614971017936543ous_on(A,C,S3,aa(fun(B,C),fun(A,C),aTP_Lamp_atn(fun(A,B),fun(fun(B,C),fun(A,C)),G),F2)) ) ) ) ).

% uniformly_continuous_on_compose
tff(fact_7475_filtermap__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),F4: fun(C,filter(B)),B4: set(C)] : filtermap(B,A,F2,aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(C),set(filter(B)),image(C,filter(B),F4),B4))) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(C),set(filter(A)),image(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ato(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F4)),B4)) ).

% filtermap_SUP
tff(fact_7476_filtermap__fun__inverse,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),F4: filter(B),G7: filter(A),F2: fun(B,A)] :
      ( filterlim(A,B,G,F4,G7)
     => ( filterlim(B,A,F2,G7,F4)
       => ( eventually(A,aa(fun(B,A),fun(A,$o),aTP_Lamp_atp(fun(A,B),fun(fun(B,A),fun(A,$o)),G),F2),G7)
         => ( filtermap(B,A,F2,F4) = G7 ) ) ) ) ).

% filtermap_fun_inverse
tff(fact_7477_eventually__filtermap,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),F4: filter(B)] :
      ( eventually(A,P,filtermap(B,A,F2,F4))
    <=> eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_akr(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2),F4) ) ).

% eventually_filtermap
tff(fact_7478_filterlim__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(A,B),F12: filter(B),G: fun(C,A),F23: filter(C)] :
      ( filterlim(A,B,F2,F12,filtermap(C,A,G,F23))
    <=> filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acn(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),F12,F23) ) ).

% filterlim_filtermap
tff(fact_7479_uniformly__continuous__on__id,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S3: set(A)] : topolo6026614971017936543ous_on(A,A,S3,aTP_Lamp_atq(A,A)) ) ).

% uniformly_continuous_on_id
tff(fact_7480_uniformly__continuous__on__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo7287701948861334536_space(B)
        & topolo7287701948861334536_space(A) )
     => ! [S3: set(A),C2: B] : topolo6026614971017936543ous_on(A,B,S3,aTP_Lamp_atr(B,fun(A,B),C2)) ) ).

% uniformly_continuous_on_const
tff(fact_7481_filtermap__ident,axiom,
    ! [A: $tType,F4: filter(A)] : filtermap(A,A,aTP_Lamp_ml(A,A),F4) = F4 ).

% filtermap_ident
tff(fact_7482_filtermap__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),F4: filter(C)] : filtermap(B,A,F2,filtermap(C,B,G,F4)) = filtermap(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_mr(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G),F4) ).

% filtermap_filtermap
tff(fact_7483_uniformly__continuous__on__Cauchy,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [S: set(A),F2: fun(A,B),X5: fun(nat,A)] :
          ( topolo6026614971017936543ous_on(A,B,S,F2)
         => ( topolo3814608138187158403Cauchy(A,X5)
           => ( ! [N: nat] : aa(set(A),$o,member(A,aa(nat,A,X5,N)),S)
             => topolo3814608138187158403Cauchy(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ats(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ) ).

% uniformly_continuous_on_Cauchy
tff(fact_7484_filtermap__nhds__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [D2: A,A2: A] : filtermap(A,A,aTP_Lamp_att(A,fun(A,A),D2),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2)) ) ).

% filtermap_nhds_shift
tff(fact_7485_filtermap__nhds__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : filtermap(A,A,uminus_uminus(A),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,uminus_uminus(A),A2)) ) ).

% filtermap_nhds_minus
tff(fact_7486_filtermap__nhds__times,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C2),topolo7230453075368039082e_nhds(A,A2)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ) ).

% filtermap_nhds_times
tff(fact_7487_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ? [N: nat,F3: fun(nat,A)] :
          ( ( A3 = aa(set(nat),set(A),image(nat,A,F3),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),N))) )
          & inj_on(nat,A,F3,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_7488_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ? [F3: fun(A,nat),N: nat] :
          ( ( aa(set(A),set(nat),image(A,nat,F3),A3) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),N)) )
          & inj_on(A,nat,F3,A3) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_7489_filtermap__at__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [D2: A,A2: A] : filtermap(A,A,aTP_Lamp_att(A,fun(A,A),D2),topolo174197925503356063within(A,A2,top_top(set(A)))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D2),top_top(set(A))) ) ).

% filtermap_at_shift
tff(fact_7490_filtermap__at__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : filtermap(A,A,uminus_uminus(A),topolo174197925503356063within(A,A2,top_top(set(A)))) = topolo174197925503356063within(A,aa(A,A,uminus_uminus(A),A2),top_top(set(A))) ) ).

% filtermap_at_minus
tff(fact_7491_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X2: nat] :
            ( aa(set(nat),$o,member(nat,X2),I5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs)) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_7492_filtermap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),F4: fun(C,filter(B)),B4: set(C)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),filtermap(B,A,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image(C,filter(B),F4),B4)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ato(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F4)),B4))) ).

% filtermap_INF
tff(fact_7493_isUCont__Cauchy,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),X5: fun(nat,A)] :
          ( topolo6026614971017936543ous_on(A,B,top_top(set(A)),F2)
         => ( topolo3814608138187158403Cauchy(A,X5)
           => topolo3814608138187158403Cauchy(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ats(fun(A,B),fun(fun(nat,A),fun(nat,B)),F2),X5)) ) ) ) ).

% isUCont_Cauchy
tff(fact_7494_summable__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X2))
         => summable(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F2),G)) ) ) ) ).

% summable_reindex
tff(fact_7495_inj__on__funpow__least,axiom,
    ! [A: $tType,Nb: nat,F2: fun(A,A),S3: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F2),S3) = S3 )
     => ( ! [M: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Nb)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F2),S3) != S3 ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_atu(fun(A,A),fun(A,fun(nat,A)),F2),S3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% inj_on_funpow_least
tff(fact_7496_at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A] : topolo174197925503356063within(A,A2,top_top(set(A))) = filtermap(A,A,aTP_Lamp_atv(A,fun(A,A),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% at_to_0
tff(fact_7497_uniformly__continuous__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S3,F2)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X: A] :
                      ( aa(set(A),$o,member(A,X),S3)
                     => ! [Xa2: A] :
                          ( aa(set(A),$o,member(A,Xa2),S3)
                         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa2,X)),D4)
                           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,Xa2),aa(A,B,F2,X))),E4) ) ) ) ) ) ) ) ).

% uniformly_continuous_on_def
tff(fact_7498_filterlim__INF__INF,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,J4: set(A),I5: set(B),F2: fun(D,C),F4: fun(B,filter(D)),G7: fun(A,filter(C))] :
      ( ! [M: A] :
          ( aa(set(A),$o,member(A,M),J4)
         => ? [X4: B] :
              ( aa(set(B),$o,member(B,X4),I5)
              & aa(filter(C),$o,aa(filter(C),fun(filter(C),$o),ord_less_eq(filter(C)),filtermap(D,C,F2,aa(B,filter(D),F4,X4))),aa(A,filter(C),G7,M)) ) )
     => filterlim(D,C,F2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image(A,filter(C),G7),J4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image(B,filter(D),F4),I5))) ) ).

% filterlim_INF_INF
tff(fact_7499_filtermap__times__pos__at__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [C2: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C2)
         => ( filtermap(A,A,aa(A,fun(A,A),times_times(A),C2),topolo174197925503356063within(A,P2,aa(A,set(A),set_ord_greaterThan(A),P2))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2),aa(A,set(A),set_ord_greaterThan(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2))) ) ) ) ).

% filtermap_times_pos_at_right
tff(fact_7500_suminf__reindex__mono,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F2),G))),suminf(real,F2)) ) ) ) ).

% suminf_reindex_mono
tff(fact_7501_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,Xb: A,F4: filter(B)] : prod_filter(A,B,aa(set(A),filter(A),principal(A),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))),F4) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),F4) ).

% prod_filter_principal_singleton
tff(fact_7502_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),Xb: B] : prod_filter(A,B,F4,aa(set(B),filter(B),principal(B),aa(set(B),set(B),insert(B,Xb),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_vd(B,fun(A,product_prod(A,B))),Xb),F4) ).

% prod_filter_principal_singleton2
tff(fact_7503_cauchy__filter__metric__filtermap,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F2: fun(B,A),F4: filter(B)] :
          ( topolo6773858410816713723filter(A,filtermap(B,A,F2,F4))
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [P5: fun(B,$o)] :
                  ( eventually(B,P5,F4)
                  & ! [X: B,Y3: B] :
                      ( ( aa(B,$o,P5,X)
                        & aa(B,$o,P5,Y3) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(B,A,F2,X),aa(B,A,F2,Y3))),E4) ) ) ) ) ) ).

% cauchy_filter_metric_filtermap
tff(fact_7504_isUCont__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,top_top(set(A)),F2)
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A,Y3: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y3)),S7)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X),aa(A,B,F2,Y3))),R5) ) ) ) ) ) ).

% isUCont_def
tff(fact_7505_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),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_7506_suminf__reindex,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F2)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X2))
         => ( ! [X2: nat] :
                ( ~ aa(set(nat),$o,member(nat,X2),aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,real,F2,X2) = zero_zero(real) ) )
           => ( suminf(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F2),G)) = suminf(real,F2) ) ) ) ) ) ).

% suminf_reindex
tff(fact_7507_pos__deriv__imp__strict__mono,axiom,
    ! [F2: fun(real,real),F6: fun(real,real)] :
      ( ! [X2: real] : has_field_derivative(real,F2,aa(real,real,F6,X2),topolo174197925503356063within(real,X2,top_top(set(real))))
     => ( ! [X2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F6,X2))
       => order_strict_mono(real,real,F2) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_7508_prod__encode__def,axiom,
    nat_prod_encode = aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),aTP_Lamp_atw(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_7509_uminus__int__def,axiom,
    uminus_uminus(int) = aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_qp(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_7510_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),B2))) ).

% le_prod_encode_1
tff(fact_7511_le__prod__encode__2,axiom,
    ! [B2: nat,A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A2),B2))) ).

% le_prod_encode_2
tff(fact_7512_filtermap__at__right__shift,axiom,
    ! [D2: real,A2: real] : filtermap(real,real,aTP_Lamp_atx(real,fun(real,real),D2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) = topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),D2),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),D2))) ).

% filtermap_at_right_shift
tff(fact_7513_at__right__to__0,axiom,
    ! [A2: real] : topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)) = filtermap(real,real,aTP_Lamp_aty(real,fun(real,real),A2),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% at_right_to_0
tff(fact_7514_at__left__minus,axiom,
    ! [A2: real] : topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2)) = filtermap(real,real,uminus_uminus(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,uminus_uminus(real),A2)))) ).

% at_left_minus
tff(fact_7515_at__right__minus,axiom,
    ! [A2: real] : topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2)) = filtermap(real,real,uminus_uminus(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),A2),aa(real,set(real),set_ord_lessThan(real),aa(real,real,uminus_uminus(real),A2)))) ).

% at_right_minus
tff(fact_7516_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Mb: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K,Mb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_7517_times__int__def,axiom,
    times_times(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qo(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_7518_minus__int__def,axiom,
    minus_minus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qy(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_7519_plus__int__def,axiom,
    plus_plus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_7520_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F2: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_atz(fun(B,list(A)),fun(B,set(A)),F2)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_7521_list__encode_Oelims,axiom,
    ! [Xb: list(nat),Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( ( ( Xb = nil(nat) )
         => ( Y != zero_zero(nat) ) )
       => ~ ! [X2: nat,Xs2: list(nat)] :
              ( ( Xb = cons(nat,X2,Xs2) )
             => ( Y != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_7522_list__bind__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F2: fun(A,list(B)),G: fun(A,list(B))] :
      ( ( Xs = Ys )
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
           => ( aa(A,list(B),F2,X2) = aa(A,list(B),G,X2) ) )
       => ( bind(A,B,Xs,F2) = bind(A,B,Ys,G) ) ) ) ).

% list_bind_cong
tff(fact_7523_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list(nat)] : nat_list_encode(cons(nat,Xb,Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_7524_has__derivative__power__int_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,Nb: int,S: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_derivative(A,A,aTP_Lamp_aua(int,fun(A,A),Nb),aa(int,fun(A,A),aTP_Lamp_aub(A,fun(int,fun(A,A)),Xb),Nb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_power_int'
tff(fact_7525_has__derivative__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(B,A),Xb: B,F6: fun(B,A),S: set(B),Nb: int] :
          ( ( aa(B,A,F2,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F6,topolo174197925503356063within(B,Xb,S))
           => has_derivative(B,A,aa(int,fun(B,A),aTP_Lamp_auc(fun(B,A),fun(int,fun(B,A)),F2),Nb),aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_aud(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),F2),Xb),F6),Nb),topolo174197925503356063within(B,Xb,S)) ) ) ) ).

% has_derivative_power_int
tff(fact_7526_power__int__1__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int] : power_int(A,one_one(A),Nb) = one_one(A) ) ).

% power_int_1_left
tff(fact_7527_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,W: num,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(num,A,numeral_numeral(A),W)),Mb) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Mb)),power_int(A,aa(num,A,numeral_numeral(A),W),Mb)) ) ).

% power_int_mult_distrib_numeral2
tff(fact_7528_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W: num,Y: A,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y),Mb) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W),Mb)),power_int(A,Y,Mb)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_7529_power__int__0__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A] : power_int(A,Xb,zero_zero(int)) = one_one(A) ) ).

% power_int_0_right
tff(fact_7530_power__int__of__nat,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A,Nb: nat] : power_int(A,Xb,aa(nat,int,semiring_1_of_nat(int),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb) ) ).

% power_int_of_nat
tff(fact_7531_power__int__mult__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: num,Nb: num] : power_int(A,power_int(A,Xb,aa(num,int,numeral_numeral(int),Mb)),aa(num,int,numeral_numeral(int),Nb)) = power_int(A,Xb,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),times_times(num),Mb),Nb))) ) ).

% power_int_mult_numeral
tff(fact_7532_power__int__minus__one__mult__self_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Mb: int,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Mb)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Mb)),B2)) = B2 ) ).

% power_int_minus_one_mult_self'
tff(fact_7533_power__int__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Mb: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Mb)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Mb)) = one_one(A) ) ).

% power_int_minus_one_mult_self
tff(fact_7534_power__int__numeral,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A,Nb: num] : power_int(A,Xb,aa(num,int,numeral_numeral(int),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% power_int_numeral
tff(fact_7535_numeral__power__int__eq__of__real__cancel__iff,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: num,Nb: int,Y: real] :
          ( ( power_int(A,aa(num,A,numeral_numeral(A),Xb),Nb) = aa(real,A,real_Vector_of_real(A),Y) )
        <=> ( power_int(real,aa(num,real,numeral_numeral(real),Xb),Nb) = Y ) ) ) ).

% numeral_power_int_eq_of_real_cancel_iff
tff(fact_7536_of__real__eq__numeral__power__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Y: real,Xb: num,Nb: int] :
          ( ( aa(real,A,real_Vector_of_real(A),Y) = power_int(A,aa(num,A,numeral_numeral(A),Xb),Nb) )
        <=> ( Y = power_int(real,aa(num,real,numeral_numeral(real),Xb),Nb) ) ) ) ).

% of_real_eq_numeral_power_int_cancel_iff
tff(fact_7537_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),Mb))),power_int(A,Xb,aa(num,int,numeral_numeral(int),Nb))) = power_int(A,Xb,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb))) ) ).

% power_int_add_numeral
tff(fact_7538_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: num,Nb: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),Mb))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),Nb))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),Mb),Nb)))),B2) ) ).

% power_int_add_numeral2
tff(fact_7539_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A2,Nb)),power_int(A,B2,Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_7540_power__int__minus__left__odd,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int,A2: A] :
          ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A2),Nb) = aa(A,A,uminus_uminus(A),power_int(A,A2,Nb)) ) ) ) ).

% power_int_minus_left_odd
tff(fact_7541_power__int__minus__left__even,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A2),Nb) = power_int(A,A2,Nb) ) ) ) ).

% power_int_minus_left_even
tff(fact_7542_power__int__numeral__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: num,Nb: num] : power_int(A,aa(num,A,numeral_numeral(A),Mb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),pow(Mb,Nb))) ) ).

% power_int_numeral_neg_numeral
tff(fact_7543_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N2: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A2,Nb)),power_int(A,A2,N2)) ) ) ) ).

% power_int_increasing
tff(fact_7544_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),power_int(A,Xb,Nb)) ) ) ).

% zero_le_power_int
tff(fact_7545_continuous__on__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),F2: fun(A,B),Nb: int] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),S3)
               => ( aa(A,B,F2,X2) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S3,aa(int,fun(A,B),aTP_Lamp_aue(fun(A,B),fun(int,fun(A,B)),F2),Nb)) ) ) ) ).

% continuous_on_power_int
tff(fact_7546_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),power_int(A,Xb,Nb)) ) ) ).

% zero_less_power_int
tff(fact_7547_power__int__mult,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: int,Nb: int] : power_int(A,Xb,aa(int,int,aa(int,fun(int,int),times_times(int),Mb),Nb)) = power_int(A,power_int(A,Xb,Mb),Nb) ) ).

% power_int_mult
tff(fact_7548_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y),Mb) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Mb)),power_int(A,Y,Mb)) ) ).

% power_int_mult_distrib
tff(fact_7549_power__int__commutes,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Nb)),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),power_int(A,Xb,Nb)) ) ).

% power_int_commutes
tff(fact_7550_power__int__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Xb),Nb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),power_int(A,Xb,Nb)) ) ).

% power_int_one_over
tff(fact_7551_power__int__divide__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Mb: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y),Mb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,Xb,Mb)),power_int(A,Y,Mb)) ) ).

% power_int_divide_distrib
tff(fact_7552_power__int__minus__one__diff__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: int,B2: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),A2),B2)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),A2)) ) ).

% power_int_minus_one_diff_commute
tff(fact_7553_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Mb: int,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Mb != Nb ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,Xb,Mb)),power_int(A,Xb,Nb)) ) ) ) ).

% power_int_diff
tff(fact_7554_power__int__minus__one__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,uminus_uminus(int),Nb)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),Nb) ) ).

% power_int_minus_one_minus
tff(fact_7555_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N2: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A2,Nb)),power_int(A,A2,N2)) ) ) ) ).

% power_int_strict_increasing
tff(fact_7556_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Mb: int] :
          power_int(A,zero_zero(A),Mb) = $ite(Mb = zero_zero(int),one_one(A),zero_zero(A)) ) ).

% power_int_0_left_If
tff(fact_7557_tendsto__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A2: B,F4: filter(A),Nb: int] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aa(int,fun(A,B),aTP_Lamp_auf(fun(A,B),fun(int,fun(A,B)),F2),Nb),topolo7230453075368039082e_nhds(B,power_int(B,A2,Nb)),F4) ) ) ) ).

% tendsto_power_int
tff(fact_7558_continuous__at__within__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,S3: set(A),F2: fun(A,B),Nb: int] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S3),F2)
         => ( ( aa(A,B,F2,A2) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S3),aa(int,fun(A,B),aTP_Lamp_aug(fun(A,B),fun(int,fun(A,B)),F2),Nb)) ) ) ) ).

% continuous_at_within_power_int
tff(fact_7559_differentiable__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xb: A,S3: set(A),Nb: int] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xb,S3))
         => ( ( aa(A,B,F2,Xb) != zero_zero(B) )
           => differentiable(A,B,aa(int,fun(A,B),aTP_Lamp_auh(fun(A,B),fun(int,fun(A,B)),F2),Nb),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% differentiable_power_int
tff(fact_7560_continuous__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F2: fun(A,B),Nb: int] :
          ( topolo3448309680560233919inuous(A,B,F4,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_aac(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F4,aa(int,fun(A,B),aTP_Lamp_aug(fun(A,B),fun(int,fun(A,B)),F2),Nb)) ) ) ) ).

% continuous_power_int
tff(fact_7561_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N2: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A2,N2)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_7562_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xb,Nb)),power_int(A,Y,Nb)) ) ) ) ) ).

% power_int_mono
tff(fact_7563_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,B2,Nb)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_7564_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),power_int(A,Xb,Nb)) ) ) ) ).

% one_le_power_int
tff(fact_7565_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),power_int(A,A2,Nb)) ) ) ) ).

% one_less_power_int
tff(fact_7566_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: int,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),Mb),Nb) != zero_zero(int) ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),plus_plus(int),Mb),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Mb)),power_int(A,Xb,Nb)) ) ) ) ).

% power_int_add
tff(fact_7567_power__int__minus__left__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( division_ring(C)
        & one(A)
        & uminus(A) )
     => ! [Xb: B,A2: C,Nb: int] :
          ( nO_MATCH(A,B,aa(A,A,uminus_uminus(A),one_one(A)),Xb)
         => ( power_int(C,aa(C,C,uminus_uminus(C),A2),Nb) = aa(C,C,aa(C,fun(C,C),times_times(C),power_int(C,aa(C,C,uminus_uminus(C),one_one(C)),Nb)),power_int(C,A2,Nb)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_7568_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,B2,Nb)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_antimono
tff(fact_7569_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A2,Nb)),power_int(A,B2,Nb)) ) ) ) ) ).

% power_int_strict_mono
tff(fact_7570_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N2: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => ( ( ( A2 != zero_zero(A) )
                  | ( N2 != zero_zero(int) )
                  | ( Nb = zero_zero(int) ) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A2,N2)),power_int(A,A2,Nb)) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_7571_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xb,Nb)),one_one(A)) ) ) ) ) ).

% power_int_le_one
tff(fact_7572_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Mb: int,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xb,Mb)),power_int(A,Xb,Nb))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Mb),Nb) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_7573_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Mb: int,Nb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,Xb,Mb)),power_int(A,Xb,Nb))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Mb),Nb) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_7574_power__int__minus__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: int] :
          power_int(A,aa(A,A,uminus_uminus(A),A2),Nb) = $ite(aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Nb),power_int(A,A2,Nb),aa(A,A,uminus_uminus(A),power_int(A,A2,Nb))) ) ).

% power_int_minus_left
tff(fact_7575_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Nb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Nb != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,aa(int,int,aa(int,fun(int,int),minus_minus(int),Nb),one_one(int)))),Xb) = power_int(A,Xb,Nb) ) ) ) ).

% power_int_minus_mult
tff(fact_7576_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Mb != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),plus_plus(int),Mb),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),power_int(A,Xb,Mb)) ) ) ) ).

% power_int_add_1'
tff(fact_7577_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Mb: int] :
          ( ( ( Xb != zero_zero(A) )
            | ( Mb != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xb,aa(int,int,aa(int,fun(int,int),plus_plus(int),Mb),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xb,Mb)),Xb) ) ) ) ).

% power_int_add_1
tff(fact_7578_power__int__def,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xb: A,Nb: int] :
          power_int(A,Xb,Nb) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(int,nat,nat2,Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Nb)))) ) ).

% power_int_def
tff(fact_7579_powr__real__of__int_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( ( Xb != zero_zero(real) )
          | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Nb) )
       => ( powr(real,Xb,ring_1_of_int(real,Nb)) = power_int(real,Xb,Nb) ) ) ) ).

% powr_real_of_int'
tff(fact_7580_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xb: A,S3: set(A),Nb: int] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xb,S3))
         => ( ( aa(A,A,F2,Xb) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_aui(fun(A,A),fun(int,fun(A,A)),F2),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),ring_1_of_int(A,Nb)),power_int(A,aa(A,A,F2,Xb),aa(int,int,aa(int,fun(int,int),minus_minus(int),Nb),one_one(int))))),D2),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_power_int
tff(fact_7581_lists__length__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] : aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_auj(set(A),fun(nat,fun(list(A),$o)),A3),Nb)) = aa(set(product_prod(list(A),A)),set(list(A)),image(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_atm(list(A),fun(A,list(A))))),product_Sigma(list(A),A,aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_aj(set(A),fun(nat,fun(list(A),$o)),A3),Nb)),aTP_Lamp_auk(set(A),fun(list(A),set(A)),A3))) ).

% lists_length_Suc_eq
tff(fact_7582_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P2: fun(A,B),I2: A] :
          ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),P2)))
         => ( groups1962203154675924110t_prod(A,B,P2,aa(set(A),set(A),insert(A,I2),I5)) = $ite(aa(set(A),$o,member(A,I2),I5),groups1962203154675924110t_prod(A,B,P2,I5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P2,I2)),groups1962203154675924110t_prod(A,B,P2,I5))) ) ) ) ).

% prod.insert'
tff(fact_7583_SigmaI,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B2: B,B4: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,A2),A3)
     => ( aa(set(B),$o,member(B,B2),aa(A,set(B),B4,A2))
       => aa(set(product_prod(A,B)),$o,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)),product_Sigma(A,B,A3,B4)) ) ) ).

% SigmaI
tff(fact_7584_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,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)),product_Sigma(A,B,A3,B4))
    <=> ( aa(set(A),$o,member(A,A2),A3)
        & aa(set(B),$o,member(B,B2),aa(A,set(B),B4,A2)) ) ) ).

% mem_Sigma_iff
tff(fact_7585_finite__SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: fun(A,set(B))] :
      ( finite_finite2(A,A3)
     => ( ! [A4: A] :
            ( aa(set(A),$o,member(A,A4),A3)
           => finite_finite2(B,aa(A,set(B),B4,A4)) )
       => finite_finite2(product_prod(A,B),product_Sigma(A,B,A3,B4)) ) ) ).

% finite_SigmaI
tff(fact_7586_Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),Q: fun(B,$o)] : aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_arx(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),P),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P),aTP_Lamp_aul(fun(B,$o),fun(A,set(B)),Q)) ).

% Collect_case_prod
tff(fact_7587_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: fun(B,A)] : groups1962203154675924110t_prod(B,A,P2,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_7588_Compl__Times__UNIV2,axiom,
    ! [B: $tType,A: $tType,A3: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aum(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A3),aTP_Lamp_aum(A,set(B))) ).

% Compl_Times_UNIV2
tff(fact_7589_Compl__Times__UNIV1,axiom,
    ! [A: $tType,B: $tType,A3: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aun(set(B),fun(A,set(B)),A3))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_auo(set(B),fun(A,set(B)),A3)) ).

% Compl_Times_UNIV1
tff(fact_7590_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A3: set(A)] : product_Sigma(A,B,A3,aTP_Lamp_aup(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_7591_Times__empty,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] :
      ( ( product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( B4 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_7592_UNIV__Times__UNIV,axiom,
    ! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aum(A,set(B))) = top_top(set(product_prod(A,B))) ).

% UNIV_Times_UNIV
tff(fact_7593_card__SigmaI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: fun(A,set(B))] :
      ( finite_finite2(A,A3)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => finite_finite2(B,aa(A,set(B),B4,X2)) )
       => ( aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A3,B4)) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_su(fun(A,set(B)),fun(A,nat),B4)),A3) ) ) ) ).

% card_SigmaI
tff(fact_7594_fst__image__times,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B)] :
      aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4))) = $ite(B4 = bot_bot(set(B)),bot_bot(set(A)),A3) ).

% fst_image_times
tff(fact_7595_snd__image__times,axiom,
    ! [B: $tType,A: $tType,A3: set(B),B4: set(A)] :
      aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A3,aTP_Lamp_rz(set(A),fun(B,set(A)),B4))) = $ite(A3 = bot_bot(set(B)),bot_bot(set(A)),B4) ).

% snd_image_times
tff(fact_7596_set__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),product(A,B,Xs,Ys)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xs),aTP_Lamp_auq(list(B),fun(A,set(B)),Ys)) ).

% set_product
tff(fact_7597_inj__on__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(A,C),A3: set(A)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F2),product_Sigma(A,B,A3,aTP_Lamp_aum(A,set(B))))
    <=> inj_on(A,C,F2,A3) ) ).

% inj_on_apfst
tff(fact_7598_inj__on__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(B,C),A3: set(B)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F2),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_aun(set(B),fun(A,set(B)),A3)))
    <=> inj_on(B,C,F2,A3) ) ).

% inj_on_apsnd
tff(fact_7599_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,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)),product_Sigma(A,B,A3,B4))
     => ~ ( aa(set(A),$o,member(A,A2),A3)
         => ~ aa(set(B),$o,member(B,B2),aa(A,set(B),B4,A2)) ) ) ).

% SigmaE2
tff(fact_7600_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,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)),product_Sigma(A,B,A3,B4))
     => aa(set(B),$o,member(B,B2),aa(A,set(B),B4,A2)) ) ).

% SigmaD2
tff(fact_7601_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,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)),product_Sigma(A,B,A3,B4))
     => aa(set(A),$o,member(A,A2),A3) ) ).

% SigmaD1
tff(fact_7602_SigmaE,axiom,
    ! [A: $tType,B: $tType,C2: product_prod(A,B),A3: set(A),B4: fun(A,set(B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),C2),product_Sigma(A,B,A3,B4))
     => ~ ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => ! [Y4: B] :
                ( aa(set(B),$o,member(B,Y4),aa(A,set(B),B4,X2))
               => ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y4) ) ) ) ) ).

% SigmaE
tff(fact_7603_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aur(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_7604_mem__Times__iff,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),A3: set(A),B4: set(B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),Xb),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4)))
    <=> ( aa(set(A),$o,member(A,aa(product_prod(A,B),A,product_fst(A,B),Xb)),A3)
        & aa(set(B),$o,member(B,aa(product_prod(A,B),B,product_snd(A,B),Xb)),B4) ) ) ).

% mem_Times_iff
tff(fact_7605_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Diff_distrib1
tff(fact_7606_Times__eq__cancel2,axiom,
    ! [A: $tType,B: $tType,Xb: A,C5: set(A),A3: set(B),B4: set(B)] :
      ( aa(set(A),$o,member(A,Xb),C5)
     => ( ( product_Sigma(B,A,A3,aTP_Lamp_rz(set(A),fun(B,set(A)),C5)) = product_Sigma(B,A,B4,aTP_Lamp_rz(set(A),fun(B,set(A)),C5)) )
      <=> ( A3 = B4 ) ) ) ).

% Times_eq_cancel2
tff(fact_7607_Sigma__cong,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(A),C5: fun(A,set(B)),D5: fun(A,set(B))] :
      ( ( A3 = B4 )
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),B4)
           => ( aa(A,set(B),C5,X2) = aa(A,set(B),D5,X2) ) )
       => ( product_Sigma(A,B,A3,C5) = product_Sigma(A,B,B4,D5) ) ) ) ).

% Sigma_cong
tff(fact_7608_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1962203154675924110t_prod(B,A,G,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_aus(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1962203154675924110t_prod(B,A,G,I5) ) ).

% prod.non_neutral'
tff(fact_7609_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4),aTP_Lamp_aun(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B4,aTP_Lamp_aun(set(B),fun(A,set(B)),C5))) ).

% Times_Diff_distrib1
tff(fact_7610_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),B4: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_aut(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A3),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I5,A3)),product_Sigma(A,B,I5,B4)) ).

% Sigma_Diff_distrib2
tff(fact_7611_finite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(B,B4)
       => finite_finite2(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4))) ) ) ).

% finite_cartesian_product
tff(fact_7612_card__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B4)) ).

% card_cartesian_product
tff(fact_7613_Times__Int__Times,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),C5: set(A),D5: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4))),product_Sigma(A,B,C5,aTP_Lamp_aun(set(B),fun(A,set(B)),D5))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),C5),aa(set(B),fun(A,set(B)),aTP_Lamp_auu(set(B),fun(set(B),fun(A,set(B))),B4),D5)) ).

% Times_Int_Times
tff(fact_7614_Sigma__Int__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),B4: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_auv(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A3),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I5,A3)),product_Sigma(A,B,I5,B4)) ).

% Sigma_Int_distrib2
tff(fact_7615_Times__Int__distrib1,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4),aTP_Lamp_aun(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B4,aTP_Lamp_aun(set(B),fun(A,set(B)),C5))) ).

% Times_Int_distrib1
tff(fact_7616_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I5: set(A),X5: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I5,X5) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X: A] :
          ( aa(set(A),$o,member(A,X),I5)
         => ( aa(A,set(B),X5,X) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_7617_finite__cartesian__product__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] :
      ( finite_finite2(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4)))
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( B4 = bot_bot(set(B)) )
        | ( finite_finite2(A,A3)
          & finite_finite2(B,B4) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_7618_finite__cartesian__productD2,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] :
      ( finite_finite2(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4)))
     => ( ( A3 != bot_bot(set(A)) )
       => finite_finite2(B,B4) ) ) ).

% finite_cartesian_productD2
tff(fact_7619_finite__cartesian__productD1,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B)] :
      ( finite_finite2(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4)))
     => ( ( B4 != bot_bot(set(B)) )
       => finite_finite2(A,A3) ) ) ).

% finite_cartesian_productD1
tff(fact_7620_finite__SigmaI2,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: fun(A,set(B))] :
      ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_auw(set(A),fun(fun(A,set(B)),fun(A,$o)),A3),B4)))
     => ( ! [A4: A] :
            ( aa(set(A),$o,member(A,A4),A3)
           => finite_finite2(B,aa(A,set(B),B4,A4)) )
       => finite_finite2(product_prod(A,B),product_Sigma(A,B,A3,B4)) ) ) ).

% finite_SigmaI2
tff(fact_7621_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B),C5: set(A),D5: set(B)] :
      ( ( product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4)) = product_Sigma(A,B,C5,aTP_Lamp_aun(set(B),fun(A,set(B)),D5)) )
    <=> ( ( ( A3 = C5 )
          & ( B4 = D5 ) )
        | ( ( ( A3 = bot_bot(set(A)) )
            | ( B4 = bot_bot(set(B)) ) )
          & ( ( C5 = bot_bot(set(A)) )
            | ( D5 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_7622_closed__Times,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),T2: set(B)] :
          ( topolo7761053866217962861closed(A,S)
         => ( topolo7761053866217962861closed(B,T2)
           => topolo7761053866217962861closed(product_prod(A,B),product_Sigma(A,B,S,aTP_Lamp_aux(set(B),fun(A,set(B)),T2))) ) ) ) ).

% closed_Times
tff(fact_7623_open__Times,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),T2: set(B)] :
          ( aa(set(A),$o,topolo1002775350975398744n_open(A),S)
         => ( aa(set(B),$o,topolo1002775350975398744n_open(B),T2)
           => aa(set(product_prod(A,B)),$o,topolo1002775350975398744n_open(product_prod(A,B)),product_Sigma(A,B,S,aTP_Lamp_aux(set(B),fun(A,set(B)),T2))) ) ) ) ).

% open_Times
tff(fact_7624_Sigma__Union,axiom,
    ! [B: $tType,A: $tType,X5: set(set(A)),B4: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X5),B4) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(set(A)),set(set(product_prod(A,B))),image(set(A),set(product_prod(A,B)),aTP_Lamp_auy(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B4)),X5)) ).

% Sigma_Union
tff(fact_7625_open__prod__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(product_prod(A,B))] :
          ( ! [X2: product_prod(A,B)] :
              ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X2),S)
             => ? [A11: set(A),B12: set(B)] :
                  ( aa(set(A),$o,topolo1002775350975398744n_open(A),A11)
                  & aa(set(B),$o,topolo1002775350975398744n_open(B),B12)
                  & aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X2),product_Sigma(A,B,A11,aTP_Lamp_aux(set(B),fun(A,set(B)),B12)))
                  & aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A11,aTP_Lamp_aux(set(B),fun(A,set(B)),B12))),S) ) )
         => aa(set(product_prod(A,B)),$o,topolo1002775350975398744n_open(product_prod(A,B)),S) ) ) ).

% open_prod_intro
tff(fact_7626_open__prod__elim,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(product_prod(A,B)),Xb: product_prod(A,B)] :
          ( aa(set(product_prod(A,B)),$o,topolo1002775350975398744n_open(product_prod(A,B)),S)
         => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),Xb),S)
           => ~ ! [A7: set(A)] :
                  ( aa(set(A),$o,topolo1002775350975398744n_open(A),A7)
                 => ! [B8: set(B)] :
                      ( aa(set(B),$o,topolo1002775350975398744n_open(B),B8)
                     => ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),Xb),product_Sigma(A,B,A7,aTP_Lamp_aux(set(B),fun(A,set(B)),B8)))
                       => ~ aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A7,aTP_Lamp_aux(set(B),fun(A,set(B)),B8))),S) ) ) ) ) ) ) ).

% open_prod_elim
tff(fact_7627_open__prod__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(product_prod(A,B))] :
          ( aa(set(product_prod(A,B)),$o,topolo1002775350975398744n_open(product_prod(A,B)),S)
        <=> ! [X: product_prod(A,B)] :
              ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X),S)
             => ? [A14: set(A)] :
                  ( aa(set(A),$o,topolo1002775350975398744n_open(A),A14)
                  & ? [B10: set(B)] :
                      ( aa(set(B),$o,topolo1002775350975398744n_open(B),B10)
                      & aa(set(product_prod(A,B)),$o,member(product_prod(A,B),X),product_Sigma(A,B,A14,aTP_Lamp_aux(set(B),fun(A,set(B)),B10)))
                      & aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A14,aTP_Lamp_aux(set(B),fun(A,set(B)),B10))),S) ) ) ) ) ) ).

% open_prod_def
tff(fact_7628_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,Xb: A,C5: set(A),A3: set(B),B4: set(B)] :
      ( aa(set(A),$o,member(A,Xb),C5)
     => ( aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A3,aTP_Lamp_rz(set(A),fun(B,set(A)),C5))),product_Sigma(B,A,B4,aTP_Lamp_rz(set(A),fun(B,set(A)),C5)))
      <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A3),B4) ) ) ).

% Times_subset_cancel2
tff(fact_7629_Collect__case__prod__Sigma,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(A,fun(B,$o))] : aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_auz(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),P),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P),aTP_Lamp_ava(fun(A,fun(B,$o)),fun(A,set(B)),Q)) ).

% Collect_case_prod_Sigma
tff(fact_7630_continuous__on__swap__args,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: set(A),B4: set(B),D2: fun(A,fun(B,C))] :
          ( topolo81223032696312382ous_on(product_prod(A,B),C,product_Sigma(A,B,A3,aTP_Lamp_aux(set(B),fun(A,set(B)),B4)),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),D2))
         => topolo81223032696312382ous_on(product_prod(B,A),C,product_Sigma(B,A,B4,aTP_Lamp_avb(set(A),fun(B,set(A)),A3)),aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),aTP_Lamp_avc(fun(A,fun(B,C)),fun(B,fun(A,C)),D2))) ) ) ).

% continuous_on_swap_args
tff(fact_7631_continuous__on__compose__Pair,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(A),B4: fun(A,set(B)),F2: fun(A,fun(B,C)),C5: set(D),G: fun(D,A),H: fun(D,B)] :
          ( topolo81223032696312382ous_on(product_prod(A,B),C,product_Sigma(A,B,A3,B4),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F2))
         => ( topolo81223032696312382ous_on(D,A,C5,G)
           => ( topolo81223032696312382ous_on(D,B,C5,H)
             => ( ! [C3: D] :
                    ( aa(set(D),$o,member(D,C3),C5)
                   => aa(set(A),$o,member(A,aa(D,A,G,C3)),A3) )
               => ( ! [C3: D] :
                      ( aa(set(D),$o,member(D,C3),C5)
                     => aa(set(B),$o,member(B,aa(D,B,H,C3)),aa(A,set(B),B4,aa(D,A,G,C3))) )
                 => topolo81223032696312382ous_on(D,C,C5,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_avd(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),F2),G),H)) ) ) ) ) ) ) ).

% continuous_on_compose_Pair
tff(fact_7632_swap__product,axiom,
    ! [A: $tType,B: $tType,A3: set(B),B4: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_vd(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A3,aTP_Lamp_rz(set(A),fun(B,set(A)),B4))) = product_Sigma(A,B,B4,aTP_Lamp_aun(set(B),fun(A,set(B)),A3)) ).

% swap_product
tff(fact_7633_card__cartesian__product__singleton,axiom,
    ! [A: $tType,B: $tType,Xb: A,A3: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))),aTP_Lamp_aun(set(B),fun(A,set(B)),A3))) = aa(set(B),nat,finite_card(B),A3) ).

% card_cartesian_product_singleton
tff(fact_7634_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),C5: set(B),B4: set(A),D5: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B4,aTP_Lamp_aun(set(B),fun(A,set(B)),D5)))
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( C5 = bot_bot(set(B)) )
        | ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C5),D5) ) ) ) ).

% times_subset_iff
tff(fact_7635_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F2: fun(C,A),G: fun(D,B),A3: set(C),B4: set(D)] : aa(set(product_prod(C,D)),set(product_prod(A,B)),image(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_ave(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F2),G))),product_Sigma(C,D,A3,aTP_Lamp_avf(set(D),fun(C,set(D)),B4))) = product_Sigma(A,B,aa(set(C),set(A),image(C,A,F2),A3),aa(set(D),fun(A,set(B)),aTP_Lamp_avg(fun(D,B),fun(set(D),fun(A,set(B))),G),B4)) ).

% image_paired_Times
tff(fact_7636_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: fun(A,set(B))] : aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A3,B4)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_auw(set(A),fun(fun(A,set(B)),fun(A,$o)),A3),B4)) ).

% fst_image_Sigma
tff(fact_7637_UN__Times__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,E5: fun(C,set(A)),F4: fun(D,set(B)),A3: set(C),B4: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(set(product_prod(A,B))),image(product_prod(C,D),set(product_prod(A,B)),aa(fun(C,fun(D,set(product_prod(A,B)))),fun(product_prod(C,D),set(product_prod(A,B))),product_case_prod(C,D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_avi(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E5),F4))),product_Sigma(C,D,A3,aTP_Lamp_avf(set(D),fun(C,set(D)),B4)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),E5),A3)),aa(set(D),fun(A,set(B)),aTP_Lamp_avj(fun(D,set(B)),fun(set(D),fun(A,set(B))),F4),B4)) ).

% UN_Times_distrib
tff(fact_7638_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,G,X2) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,S) = groups1962203154675924110t_prod(A,B,G,T2) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_7639_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,G,X2) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,T2) = groups1962203154675924110t_prod(A,B,G,S) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_7640_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [I3: A] :
                ( aa(set(A),$o,member(A,I3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,H,I3) = one_one(B) ) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S)
                 => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
             => ( groups1962203154675924110t_prod(A,B,G,S) = groups1962203154675924110t_prod(A,B,H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_7641_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T2)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S))
               => ( aa(A,B,G,X2) = one_one(B) ) )
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),S)
                 => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
             => ( groups1962203154675924110t_prod(A,B,G,T2) = groups1962203154675924110t_prod(A,B,H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_7642_sum_Ocartesian__product,axiom,
    ! [A: $tType,B: $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_by(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B4)),A3) = aa(set(product_prod(B,C)),A,groups7311177749621191930dd_sum(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A3,aTP_Lamp_avk(set(C),fun(B,set(C)),B4))) ) ).

% sum.cartesian_product
tff(fact_7643_prod_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B4: set(C),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(C),fun(B,A),aTP_Lamp_hi(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B4)),A3) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7121269368397514597t_prod(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A3,aTP_Lamp_avk(set(C),fun(B,set(C)),B4))) ) ).

% prod.cartesian_product
tff(fact_7644_principal__prod__principal,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] : prod_filter(A,B,aa(set(A),filter(A),principal(A),A3),aa(set(B),filter(B),principal(B),B4)) = aa(set(product_prod(A,B)),filter(product_prod(A,B)),principal(product_prod(A,B)),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4))) ).

% principal_prod_principal
tff(fact_7645_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aur(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_7646_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: fun(B,A),I5: set(B)] :
          groups1962203154675924110t_prod(B,A,P2,I5) = $ite(finite_finite2(B,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_aus(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P2),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_aus(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),one_one(A)) ) ).

% prod.G_def
tff(fact_7647_snd__image__Sigma,axiom,
    ! [A: $tType,B: $tType,A3: set(B),B4: fun(B,set(A))] : aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A3,B4)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) ).

% snd_image_Sigma
tff(fact_7648_subset__fst__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B),S: set(product_prod(A,B)),Y: B] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4))),S)
     => ( aa(set(B),$o,member(B,Y),B4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),S)) ) ) ).

% subset_fst_imageI
tff(fact_7649_subset__snd__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B),S: set(product_prod(A,B)),Xb: A] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4))),S)
     => ( aa(set(A),$o,member(A,Xb),A3)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B4),aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),S)) ) ) ).

% subset_snd_imageI
tff(fact_7650_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A3: set(product_prod(A,B))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A3),aTP_Lamp_avl(set(product_prod(A,B)),fun(A,set(B)),A3))) ).

% subset_fst_snd
tff(fact_7651_Sigma__interval__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [A3: set(A),V: fun(A,B),W: B] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_avm(fun(A,B),fun(A,set(B)),V))),product_Sigma(A,B,A3,aa(B,fun(A,set(B)),aTP_Lamp_avn(fun(A,B),fun(B,fun(A,set(B))),V),W))) = bot_bot(set(product_prod(A,B))) ) ).

% Sigma_interval_disjoint
tff(fact_7652_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: fun(A,set(B))] : product_Sigma(A,B,A3,B4) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(A),set(set(product_prod(A,B))),image(A,set(product_prod(A,B)),aTP_Lamp_avp(fun(A,set(B)),fun(A,set(product_prod(A,B))),B4)),A3)) ).

% Sigma_def
tff(fact_7653_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B)] :
    ? [F3: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5)))
      & aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),aa(set(A),set(product_prod(B,A)),image(A,product_prod(B,A),F3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5)))),product_Sigma(B,A,I5,A3)) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_7654_sum_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),B4: fun(A,set(B)),G: fun(A,fun(B,C))] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => finite_finite2(B,aa(A,set(B),B4,X2)) )
           => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_avq(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B4),G)),A3) = aa(set(product_prod(A,B)),C,groups7311177749621191930dd_sum(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G)),product_Sigma(A,B,A3,B4)) ) ) ) ) ).

% sum.Sigma
tff(fact_7655_prod_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),B4: fun(A,set(B)),G: fun(A,fun(B,C))] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A] :
                ( aa(set(A),$o,member(A,X2),A3)
               => finite_finite2(B,aa(A,set(B),B4,X2)) )
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_avr(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B4),G)),A3) = aa(set(product_prod(A,B)),C,aa(fun(product_prod(A,B),C),fun(set(product_prod(A,B)),C),groups7121269368397514597t_prod(product_prod(A,B),C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G)),product_Sigma(A,B,A3,B4)) ) ) ) ) ).

% prod.Sigma
tff(fact_7656_uniformly__continuous__on__uniformity,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S3,F2)
        <=> filterlim(product_prod(A,A),product_prod(B,B),aa(fun(A,fun(A,product_prod(B,B))),fun(product_prod(A,A),product_prod(B,B)),product_case_prod(A,A,product_prod(B,B)),aTP_Lamp_avs(fun(A,B),fun(A,fun(A,product_prod(B,B))),F2)),topolo7806501430040627800ormity(B),aa(filter(product_prod(A,A)),filter(product_prod(A,A)),aa(filter(product_prod(A,A)),fun(filter(product_prod(A,A)),filter(product_prod(A,A))),inf_inf(filter(product_prod(A,A))),topolo7806501430040627800ormity(A)),aa(set(product_prod(A,A)),filter(product_prod(A,A)),principal(product_prod(A,A)),product_Sigma(A,A,S3,aTP_Lamp_avt(set(A),fun(A,set(A)),S3))))) ) ) ).

% uniformly_continuous_on_uniformity
tff(fact_7657_Restr__subset,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),R: set(product_prod(A,A))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,B4,aTP_Lamp_avu(set(A),fun(A,set(A)),B4)))),product_Sigma(A,A,A3,aTP_Lamp_avu(set(A),fun(A,set(A)),A3))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A3,aTP_Lamp_avu(set(A),fun(A,set(A)),A3))) ) ) ).

% Restr_subset
tff(fact_7658_infinite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(B)] :
      ( ~ finite_finite2(A,A3)
     => ( ~ finite_finite2(B,B4)
       => ~ finite_finite2(product_prod(A,B),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4))) ) ) ).

% infinite_cartesian_product
tff(fact_7659_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_jc(nat,fun(nat,fun(nat,$o)),Mb))) = product_Sigma(nat,nat,aa(nat,set(nat),set_ord_atMost(nat),Mb),aTP_Lamp_avv(nat,fun(nat,set(nat)),Mb)) ).

% pairs_le_eq_Sigma
tff(fact_7660_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R: set(product_prod(A,A)),As2: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R,As2)
        <=> ! [I4: A,J3: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I4),J3)),R)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,As2,I4)),aa(A,B,As2,J3)) ) ) ) ).

% relChain_def
tff(fact_7661_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat))) ).

% natLess_def
tff(fact_7662_relpow__finite__bounded1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K: nat] :
      ( finite_finite2(product_prod(A,A),R2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_avw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_avx(set(product_prod(A,A)),fun(nat,$o),R2))))) ) ) ).

% relpow_finite_bounded1
tff(fact_7663_Id__on__def,axiom,
    ! [A: $tType,A3: set(A)] : id_on(A,A3) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(A),set(set(product_prod(A,A))),image(A,set(product_prod(A,A)),aTP_Lamp_avy(A,set(product_prod(A,A)))),A3)) ).

% Id_on_def
tff(fact_7664_relpow__1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),one_one(nat)),R2) = R2 ).

% relpow_1
tff(fact_7665_Id__onI,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( aa(set(A),$o,member(A,A2),A3)
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2)),id_on(A,A3)) ) ).

% Id_onI
tff(fact_7666_finite__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Nb: nat] :
      ( finite_finite2(product_prod(A,A),R2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => finite_finite2(product_prod(A,A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ) ) ).

% finite_relpow
tff(fact_7667_Id__onE,axiom,
    ! [A: $tType,C2: product_prod(A,A),A3: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),C2),id_on(A,A3))
     => ~ ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2) ) ) ) ).

% Id_onE
tff(fact_7668_Id__on__eqI,axiom,
    ! [A: $tType,A2: A,B2: A,A3: set(A)] :
      ( ( A2 = B2 )
     => ( aa(set(A),$o,member(A,A2),A3)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),id_on(A,A3)) ) ) ).

% Id_on_eqI
tff(fact_7669_Id__on__iff,axiom,
    ! [A: $tType,Xb: A,Y: A,A3: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),id_on(A,A3))
    <=> ( ( Xb = Y )
        & aa(set(A),$o,member(A,Xb),A3) ) ) ).

% Id_on_iff
tff(fact_7670_relpow__Suc__D2_H,axiom,
    ! [A: $tType,Nb: nat,R2: set(product_prod(A,A)),X4: A,Y5: A,Z2: A] :
      ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
        & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z2)),R2) )
     => ? [W2: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),W2)),R2)
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W2),Z2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ) ) ).

% relpow_Suc_D2'
tff(fact_7671_relpow__0__E,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2))
     => ( Xb = Y ) ) ).

% relpow_0_E
tff(fact_7672_relpow__0__I,axiom,
    ! [A: $tType,Xb: A,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Xb)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2)) ).

% relpow_0_I
tff(fact_7673_relpow__Suc__I2,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),Z: A,Nb: nat] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2)) ) ) ).

% relpow_Suc_I2
tff(fact_7674_relpow__Suc__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2))
     => ~ ! [Y4: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4)),R2)
           => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ) ) ).

% relpow_Suc_E2
tff(fact_7675_relpow__Suc__D2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2))
     => ? [Y4: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4)),R2)
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ) ) ).

% relpow_Suc_D2
tff(fact_7676_relpow__Suc__I,axiom,
    ! [A: $tType,Xb: A,Y: A,Nb: nat,R2: set(product_prod(A,A)),Z: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),R2)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2)) ) ) ).

% relpow_Suc_I
tff(fact_7677_relpow__Suc__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2))
     => ~ ! [Y4: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
           => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),R2) ) ) ).

% relpow_Suc_E
tff(fact_7678_relpowp__relpow__eq,axiom,
    ! [A: $tType,Nb: nat,R2: set(product_prod(A,A)),X4: A,Xa: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Nb),aTP_Lamp_avz(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X4),Xa)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ) ).

% relpowp_relpow_eq
tff(fact_7679_Id__on__def_H,axiom,
    ! [A: $tType,A3: fun(A,$o)] : id_on(A,aa(fun(A,$o),set(A),collect(A),A3)) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_awa(fun(A,$o),fun(A,fun(A,$o)),A3))) ).

% Id_on_def'
tff(fact_7680_relpow__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y4: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4)),R2)
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M),R2)) ) ) ) ) ).

% relpow_E2
tff(fact_7681_relpow__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y4: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M),R2))
               => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),R2) ) ) ) ) ).

% relpow_E
tff(fact_7682_relpow__empty,axiom,
    ! [A: $tType,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_7683_Id__on__subset__Times,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),id_on(A,A3)),product_Sigma(A,A,A3,aTP_Lamp_avu(set(A),fun(A,set(A)),A3))) ).

% Id_on_subset_Times
tff(fact_7684_relpow__fun__conv,axiom,
    ! [A: $tType,A2: A,B2: A,Nb: nat,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2))
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = A2 )
          & ( aa(nat,A,F5,Nb) = B2 )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Nb)
             => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F5,I4)),aa(nat,A,F5,aa(nat,nat,suc,I4)))),R2) ) ) ) ).

% relpow_fun_conv
tff(fact_7685_relpow__finite__bounded,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),K: nat] :
      ( finite_finite2(product_prod(A,A),R2)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_avw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_awb(set(product_prod(A,A)),fun(nat,$o),R2))))) ) ).

% relpow_finite_bounded
tff(fact_7686_ntrancl__def,axiom,
    ! [A: $tType,Nb: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,Nb,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_avw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_awc(nat,fun(nat,$o),Nb)))) ).

% ntrancl_def
tff(fact_7687_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( finite_finite2(product_prod(A,A),R2)
     => ( transitive_trancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_avw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_avx(set(product_prod(A,A)),fun(nat,$o),R2)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_7688_trancl__power,axiom,
    ! [A: $tType,P2: product_prod(A,A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),P2),transitive_trancl(A,R2))
    <=> ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),P2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N4),R2)) ) ) ).

% trancl_power
tff(fact_7689_trancl__set__ntrancl,axiom,
    ! [A: $tType,Xs: list(product_prod(A,A))] : transitive_trancl(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs))),one_one(nat)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) ).

% trancl_set_ntrancl
tff(fact_7690_finite__trancl__ntranl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( finite_finite2(product_prod(A,A),R2)
     => ( transitive_trancl(A,R2) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),R2)),one_one(nat)),R2) ) ) ).

% finite_trancl_ntranl
tff(fact_7691_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_trancl(product_prod(A,B),R))
     => ( ! [A4: A,B3: B] :
            ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))),R)
           => aa(B,$o,aa(A,fun(B,$o),P,A4),B3) )
       => ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
              ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))),transitive_trancl(product_prod(A,B),R))
             => ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R)
               => ( aa(B,$o,aa(A,fun(B,$o),P,A4),B3)
                 => aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_7692_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),R)
       => ~ ! [B3: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B3)),transitive_trancl(A,R))
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A22)),R) ) ) ) ).

% trancl.cases
tff(fact_7693_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R))
    <=> ( ? [A5: A,B5: A] :
            ( ( A1 = A5 )
            & ( A22 = B5 )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B5)),R) )
        | ? [A5: A,B5: A,C4: A] :
            ( ( A1 = A5 )
            & ( A22 = C4 )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B5)),transitive_trancl(A,R))
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),C4)),R) ) ) ) ).

% trancl.simps
tff(fact_7694_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R)
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R)) ) ).

% trancl.r_into_trancl
tff(fact_7695_tranclE,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R)
       => ~ ! [C3: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3)),transitive_trancl(A,R))
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C3),B2)),R) ) ) ) ).

% tranclE
tff(fact_7696_trancl__trans,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),Z: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),transitive_trancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),transitive_trancl(A,R))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),transitive_trancl(A,R)) ) ) ).

% trancl_trans
tff(fact_7697_trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),P: fun(A,$o)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R))
     => ( ! [Y4: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y4)),R)
           => aa(A,$o,P,Y4) )
       => ( ! [Y4: A,Z4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y4)),transitive_trancl(A,R))
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R)
               => ( aa(A,$o,P,Y4)
                 => aa(A,$o,P,Z4) ) ) )
         => aa(A,$o,P,B2) ) ) ) ).

% trancl_induct
tff(fact_7698_r__r__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R2)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R2)) ) ) ).

% r_r_into_trancl
tff(fact_7699_converse__tranclE,axiom,
    ! [A: $tType,Xb: A,Z: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),transitive_trancl(A,R))
     => ( ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),R)
       => ~ ! [Y4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4)),R)
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),transitive_trancl(A,R)) ) ) ) ).

% converse_tranclE
tff(fact_7700_irrefl__trancl__rD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xb: A,Y: A] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),transitive_trancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),R)
       => ( Xb != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_7701_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_7702_trancl__into__trancl2,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_trancl(A,R))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R)) ) ) ).

% trancl_into_trancl2
tff(fact_7703_trancl__trans__induct,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),P: fun(A,fun(A,$o))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),transitive_trancl(A,R))
     => ( ! [X2: A,Y4: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4)),R)
           => aa(A,$o,aa(A,fun(A,$o),P,X2),Y4) )
       => ( ! [X2: A,Y4: A,Z4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4)),transitive_trancl(A,R))
             => ( aa(A,$o,aa(A,fun(A,$o),P,X2),Y4)
               => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),transitive_trancl(A,R))
                 => ( aa(A,$o,aa(A,fun(A,$o),P,Y4),Z4)
                   => aa(A,$o,aa(A,fun(A,$o),P,X2),Z4) ) ) ) )
         => aa(A,$o,aa(A,fun(A,$o),P,Xb),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_7704_converse__trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),P: fun(A,$o)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R))
     => ( ! [Y4: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),B2)),R)
           => aa(A,$o,P,Y4) )
       => ( ! [Y4: A,Z4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),B2)),transitive_trancl(A,R))
               => ( aa(A,$o,P,Z4)
                 => aa(A,$o,P,Y4) ) ) )
         => aa(A,$o,P,A2) ) ) ) ).

% converse_trancl_induct
tff(fact_7705_trancl__subset__Sigma,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A3,aTP_Lamp_avu(set(A),fun(A,set(A)),A3)))
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R)),product_Sigma(A,A,A3,aTP_Lamp_avu(set(A),fun(A,set(A)),A3))) ) ).

% trancl_subset_Sigma
tff(fact_7706_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( finite_finite2(product_prod(A,A),R2)
     => ( transitive_rtrancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_avw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_awb(set(product_prod(A,A)),fun(nat,$o),R2)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_7707_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F2: fun(A,B),B4: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),bNF_Gr(A,B,A3,F2)),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4)))
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),B4) ) ).

% Gr_incl
tff(fact_7708_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R)) ) ) ).

% trancl_rtrancl_trancl
tff(fact_7709_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),Z: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),transitive_rtrancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),transitive_trancl(A,R))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),transitive_trancl(A,R)) ) ) ).

% rtrancl_trancl_trancl
tff(fact_7710_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R)) ) ) ).

% rtrancl_into_trancl2
tff(fact_7711_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_trancl(A,R)) ) ) ).

% rtrancl_into_trancl1
tff(fact_7712_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),transitive_rtrancl(A,R2))
    <=> ( ( Xb = Y )
        | ( ( Xb != Y )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),transitive_trancl(A,R2)) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_7713_trancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R)) ) ).

% trancl_into_rtrancl
tff(fact_7714_tranclD2,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),transitive_trancl(A,R2))
     => ? [Z4: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z4)),transitive_rtrancl(A,R2))
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Y)),R2) ) ) ).

% tranclD2
tff(fact_7715_rtranclD,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R2))
     => ( ( A2 = B2 )
        | ( ( A2 != B2 )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_trancl(A,R2)) ) ) ) ).

% rtranclD
tff(fact_7716_tranclD,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),transitive_trancl(A,R2))
     => ? [Z4: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z4)),R2)
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Y)),transitive_rtrancl(A,R2)) ) ) ).

% tranclD
tff(fact_7717_GrD1,axiom,
    ! [B: $tType,A: $tType,Xb: A,Fx: B,A3: set(A),F2: fun(A,B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Fx)),bNF_Gr(A,B,A3,F2))
     => aa(set(A),$o,member(A,Xb),A3) ) ).

% GrD1
tff(fact_7718_GrD2,axiom,
    ! [A: $tType,B: $tType,Xb: A,Fx: B,A3: set(A),F2: fun(A,B)] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Fx)),bNF_Gr(A,B,A3,F2))
     => ( aa(A,B,F2,Xb) = Fx ) ) ).

% GrD2
tff(fact_7719_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Bx),By)
       => ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
              ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R)
             => ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R))
               => ( aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba)
                 => aa(B,$o,aa(A,fun(B,$o),P,A4),B3) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_7720_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xaa: A,Xb: B,Za: A,Zb: B,R: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xaa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xaa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb) )
       => ~ ! [A4: A,B3: B] :
              ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xaa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))),R)
             => ~ aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za),Zb))),transitive_rtrancl(product_prod(A,B),R)) ) ) ) ).

% converse_rtranclE2
tff(fact_7721_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay)
       => ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
              ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))),transitive_rtrancl(product_prod(A,B),R))
             => ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R)
               => ( aa(B,$o,aa(A,fun(B,$o),P,A4),B3)
                 => aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Bx),By) ) ) ) ).

% rtrancl_induct2
tff(fact_7722_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R)
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),transitive_rtrancl(A,R))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_rtrancl(A,R)) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_7723_converse__rtrancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),P: fun(A,$o)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R))
     => ( aa(A,$o,P,B2)
       => ( ! [Y4: A,Z4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R)
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),B2)),transitive_rtrancl(A,R))
               => ( aa(A,$o,P,Z4)
                 => aa(A,$o,P,Y4) ) ) )
         => aa(A,$o,P,A2) ) ) ) ).

% converse_rtrancl_induct
tff(fact_7724_converse__rtranclE,axiom,
    ! [A: $tType,Xb: A,Z: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),transitive_rtrancl(A,R))
     => ( ( Xb != Z )
       => ~ ! [Y4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4)),R)
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),transitive_rtrancl(A,R)) ) ) ) ).

% converse_rtranclE
tff(fact_7725_rtrancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),P: fun(A,$o)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R))
     => ( aa(A,$o,P,A2)
       => ( ! [Y4: A,Z4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y4)),transitive_rtrancl(A,R))
             => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R)
               => ( aa(A,$o,P,Y4)
                 => aa(A,$o,P,Z4) ) ) )
         => aa(A,$o,P,B2) ) ) ) ).

% rtrancl_induct
tff(fact_7726_rtrancl__trans,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),Z: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),transitive_rtrancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z)),transitive_rtrancl(A,R))
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z)),transitive_rtrancl(A,R)) ) ) ).

% rtrancl_trans
tff(fact_7727_rtranclE,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R))
     => ( ( A2 != B2 )
       => ~ ! [Y4: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y4)),transitive_rtrancl(A,R))
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),B2)),R) ) ) ) ).

% rtranclE
tff(fact_7728_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C2: A] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2)),R)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2)),transitive_rtrancl(A,R)) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_7729_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A2: A,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2)),transitive_rtrancl(A,R)) ).

% rtrancl.rtrancl_refl
tff(fact_7730_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R))
    <=> ( ? [A5: A] :
            ( ( A1 = A5 )
            & ( A22 = A5 ) )
        | ? [A5: A,B5: A,C4: A] :
            ( ( A1 = A5 )
            & ( A22 = C4 )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B5)),transitive_rtrancl(A,R))
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),C4)),R) ) ) ) ).

% rtrancl.simps
tff(fact_7731_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R))
     => ( ( A22 != A1 )
       => ~ ! [B3: A] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B3)),transitive_rtrancl(A,R))
             => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A22)),R) ) ) ) ).

% rtrancl.cases
tff(fact_7732_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),A3: set(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,R))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A3,aTP_Lamp_avu(set(A),fun(A,set(A)),A3)))
       => ( ( A2 = B2 )
          | aa(set(A),$o,member(A,A2),A3) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_7733_rtrancl__is__UN__relpow,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : transitive_rtrancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_avw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),top_top(set(nat)))) ).

% rtrancl_is_UN_relpow
tff(fact_7734_rtrancl__imp__UN__relpow,axiom,
    ! [A: $tType,P2: product_prod(A,A),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),P2),transitive_rtrancl(A,R2))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),P2),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_avw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2)),top_top(set(nat))))) ) ).

% rtrancl_imp_UN_relpow
tff(fact_7735_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))) = remove1(A,Xb,linord4507533701916653071of_set(A,A3)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_7736_pow_Osimps_I3_J,axiom,
    ! [Xb: num,Y: num] : pow(Xb,aa(num,num,bit1,Y)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(Xb,Y))),Xb) ).

% pow.simps(3)
tff(fact_7737_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( aa(list(A),set(A),set2(A),linord4507533701916653071of_set(A,A3)) = A3 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_7738_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(list(A),nat,size_size(list(A)),linord4507533701916653071of_set(A,A3)) = aa(set(A),nat,finite_card(A),A3) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_7739_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,aa(nat,set(nat),set_ord_lessThan(nat),K)),cons(nat,K,nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_7740_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,aa(nat,set(nat),set_ord_atMost(nat),K)),cons(nat,aa(nat,nat,suc,K),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_7741_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_7742_sqr_Osimps_I2_J,axiom,
    ! [Nb: num] : sqr(aa(num,num,bit0,Nb)) = aa(num,num,bit0,aa(num,num,bit0,sqr(Nb))) ).

% sqr.simps(2)
tff(fact_7743_sqr__conv__mult,axiom,
    ! [Xb: num] : sqr(Xb) = aa(num,num,aa(num,fun(num,num),times_times(num),Xb),Xb) ).

% sqr_conv_mult
tff(fact_7744_numeral__sqr,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),sqr(K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),K)) ) ).

% numeral_sqr
tff(fact_7745_pow_Osimps_I2_J,axiom,
    ! [Xb: num,Y: num] : pow(Xb,aa(num,num,bit0,Y)) = sqr(pow(Xb,Y)) ).

% pow.simps(2)
tff(fact_7746_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,I2)),J)
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I2,J)) = cons(nat,aa(nat,nat,suc,I2),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_7747_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I2)),J)
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I2,J)) = cons(nat,aa(nat,nat,suc,I2),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_7748_sqr_Osimps_I3_J,axiom,
    ! [Nb: num] : sqr(aa(num,num,bit1,Nb)) = aa(num,num,bit1,aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(Nb)),Nb))) ).

% sqr.simps(3)
tff(fact_7749_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I2))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I2,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_7750_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,I2)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I2,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_7751_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),insert(A,Xb),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Xb),linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_7752_open__generated__order,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ( topolo1002775350975398744n_open(A) = topolo8378437560675496660pology(A,aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(A),set(set(A)),image(A,set(A),set_ord_lessThan(A)),top_top(set(A)))),aa(set(A),set(set(A)),image(A,set(A),set_ord_greaterThan(A)),top_top(set(A))))) ) ) ).

% open_generated_order
tff(fact_7753_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% sup.bounded_iff
tff(fact_7754_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% le_sup_iff
tff(fact_7755_Un__Diff__cancel,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) ).

% Un_Diff_cancel
tff(fact_7756_Un__Diff__cancel2,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)),A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),A3) ).

% Un_Diff_cancel2
tff(fact_7757_set__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),append(A,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_append
tff(fact_7758_length__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_insort
tff(fact_7759_Compl__Diff__eq,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),B4) ).

% Compl_Diff_eq
tff(fact_7760_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),F2: fun(B,A),G: fun(B,A),S: set(B)] : aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_awd(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F2),G)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,$o),set(B),collect(B),P)))),aa(set(B),set(A),image(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_awe(fun(B,$o),fun(B,$o),P))))) ).

% if_image_distrib
tff(fact_7761_UN__Un,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,set(A)),A3: set(B),B4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),M6),A3))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),M6),B4))) ).

% UN_Un
tff(fact_7762_bdd__above__image__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [F2: fun(B,A),G: fun(B,A),A3: set(B)] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_awf(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3))
        <=> ( condit941137186595557371_above(A,aa(set(B),set(A),image(B,A,F2),A3))
            & condit941137186595557371_above(A,aa(set(B),set(A),image(B,A,G),A3)) ) ) ) ).

% bdd_above_image_sup
tff(fact_7763_UN__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_awg(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5)) = $ite(C5 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5)))) ).

% UN_simps(3)
tff(fact_7764_UN__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),B4: set(A),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_awh(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C5)) = $ite(C5 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),B4)) ).

% UN_simps(2)
tff(fact_7765_UN__insert,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A2: B,A3: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),aa(set(B),set(B),insert(B,A2),A3))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B4,A2)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3))) ).

% UN_insert
tff(fact_7766_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ~ aa(set(A),$o,member(A,Xb),A3)
           => ( linord4507533701916653071of_set(A,aa(set(A),set(A),insert(A,Xb),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Xb),linord4507533701916653071of_set(A,A3)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_7767_Times__Un__distrib1,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4),aTP_Lamp_aun(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B4,aTP_Lamp_aun(set(B),fun(A,set(B)),C5))) ).

% Times_Un_distrib1
tff(fact_7768_Sigma__Un__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),B4: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_awi(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A3),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I5,A3)),product_Sigma(A,B,I5,B4)) ).

% Sigma_Un_distrib2
tff(fact_7769_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or3652927894154168847AtMost(A,Mb,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_7770_insort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Y: A,Ys: list(A)] :
          aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),cons(A,Y,Ys)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xb)),aa(A,B,F2,Y)),cons(A,Xb,cons(A,Y,Ys)),cons(A,Y,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Ys))) ) ).

% insort_key.simps(2)
tff(fact_7771_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_7772_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or3652927894154168847AtMost(A,Mb,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_7773_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI2
tff(fact_7774_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI1
tff(fact_7775_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_7776_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb_iff1
tff(fact_7777_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded2
tff(fact_7778_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded1
tff(fact_7779_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.order_iff
tff(fact_7780_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2) ) ) ) ).

% sup.boundedI
tff(fact_7781_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% sup.boundedE
tff(fact_7782_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_7783_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = Xb ) ) ) ).

% sup_absorb1
tff(fact_7784_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_7785_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb1
tff(fact_7786_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X2: A,Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F2,X2),Y4))
         => ( ! [X2: A,Y4: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),aa(A,A,aa(A,fun(A,A),F2,X2),Y4))
           => ( ! [X2: A,Y4: A,Z4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z4),X2)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,Y4),Z4)),X2) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F2,Xb),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_7787_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% sup.orderI
tff(fact_7788_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.orderE
tff(fact_7789_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_7790_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),Xb) ) ) ) ).

% sup_least
tff(fact_7791_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)) ) ) ) ).

% sup_mono
tff(fact_7792_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_7793_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI2
tff(fact_7794_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI1
tff(fact_7795_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% sup_ge2
tff(fact_7796_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% sup_ge1
tff(fact_7797_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xb) ) ) ) ).

% le_supI
tff(fact_7798_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xb)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xb) ) ) ) ).

% le_supE
tff(fact_7799_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% inf_sup_ord(3)
tff(fact_7800_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)) ) ).

% inf_sup_ord(4)
tff(fact_7801_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_7802_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_7803_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] : aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Y)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Xb)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Xb)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_7804_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F2: fun(A,B),A3: A,B4: A] :
          ( order_mono(A,B,F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A3)),aa(A,B,F2,B4))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B4))) ) ) ).

% mono_sup
tff(fact_7805_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_7806_Un__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_awj(set(A),fun(set(A),fun(A,$o)),A3),B4)) ).

% Un_def
tff(fact_7807_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aq(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_disj_eq
tff(fact_7808_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Xb),Xs)) ) ).

% insort_left_comm
tff(fact_7809_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI2
tff(fact_7810_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI1
tff(fact_7811_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_7812_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C2),A2) ) ) ) ).

% sup.strict_boundedE
tff(fact_7813_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_7814_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb3
tff(fact_7815_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI2
tff(fact_7816_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xb: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI1
tff(fact_7817_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_7818_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_7819_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_7820_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B),A2: A] :
          ( ! [X2: A] :
              ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A2)),aa(A,B,F2,X2)) )
         => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),A2),Xs) = cons(A,A2,Xs) ) ) ) ).

% insort_is_Cons
tff(fact_7821_Un__Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)) = A3 ).

% Un_Diff_Int
tff(fact_7822_Int__Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)) = A3 ).

% Int_Diff_Un
tff(fact_7823_Diff__Int,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C5)) ).

% Diff_Int
tff(fact_7824_Diff__Un,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C5)) ).

% Diff_Un
tff(fact_7825_Un__Diff,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),C5)) ).

% Un_Diff
tff(fact_7826_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_alf(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_imp_eq
tff(fact_7827_insert__def,axiom,
    ! [A: $tType,A2: A,B4: set(A)] : aa(set(A),set(A),insert(A,A2),B4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ek(A,fun(A,$o),A2))),B4) ).

% insert_def
tff(fact_7828_set__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Xs)) = aa(set(A),set(A),insert(A,Xb),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_key
tff(fact_7829_distinct__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xb: A,Xs: list(A)] :
          ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F2),Xb),Xs))
        <=> ( ~ aa(set(A),$o,member(A,Xb),aa(list(A),set(A),set2(A),Xs))
            & distinct(A,Xs) ) ) ) ).

% distinct_insort
tff(fact_7830_set__shuffles,axiom,
    ! [A: $tType,Zs2: list(A),Xs: list(A),Ys: list(A)] :
      ( aa(set(list(A)),$o,member(list(A),Zs2),shuffles(A,Xs,Ys))
     => ( aa(list(A),set(A),set2(A),Zs2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ) ) ).

% set_shuffles
tff(fact_7831_continuous__on__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(A),Ta: set(A),F2: fun(A,B),G: fun(A,B),P: fun(A,$o)] :
          ( topolo7761053866217962861closed(A,S3)
         => ( topolo7761053866217962861closed(A,Ta)
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ( topolo81223032696312382ous_on(A,B,Ta,G)
               => ( ! [X2: A] :
                      ( ( ( aa(set(A),$o,member(A,X2),S3)
                          & ~ aa(A,$o,P,X2) )
                        | ( aa(set(A),$o,member(A,X2),Ta)
                          & aa(A,$o,P,X2) ) )
                     => ( aa(A,B,F2,X2) = aa(A,B,G,X2) ) )
                 => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S3),Ta),aa(fun(A,$o),fun(A,B),aa(fun(A,B),fun(fun(A,$o),fun(A,B)),aTP_Lamp_awk(fun(A,B),fun(fun(A,B),fun(fun(A,$o),fun(A,B))),F2),G),P)) ) ) ) ) ) ) ).

% continuous_on_cases
tff(fact_7832_continuous__on__If,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(A),Ta: set(A),F2: fun(A,B),G: fun(A,B),P: fun(A,$o)] :
          ( topolo7761053866217962861closed(A,S3)
         => ( topolo7761053866217962861closed(A,Ta)
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ( topolo81223032696312382ous_on(A,B,Ta,G)
               => ( ! [X2: A] :
                      ( aa(set(A),$o,member(A,X2),S3)
                     => ( ~ aa(A,$o,P,X2)
                       => ( aa(A,B,F2,X2) = aa(A,B,G,X2) ) ) )
                 => ( ! [X2: A] :
                        ( aa(set(A),$o,member(A,X2),Ta)
                       => ( aa(A,$o,P,X2)
                         => ( aa(A,B,F2,X2) = aa(A,B,G,X2) ) ) )
                   => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S3),Ta),aa(fun(A,$o),fun(A,B),aa(fun(A,B),fun(fun(A,$o),fun(A,B)),aTP_Lamp_awk(fun(A,B),fun(fun(A,B),fun(fun(A,$o),fun(A,B))),F2),G),P)) ) ) ) ) ) ) ) ).

% continuous_on_If
tff(fact_7833_card__Un__le,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4))) ).

% card_Un_le
tff(fact_7834_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% sup_shunt
tff(fact_7835_Diff__partition,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)) = B4 ) ) ).

% Diff_partition
tff(fact_7836_Diff__subset__conv,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4)),C5)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),C5)) ) ).

% Diff_subset_conv
tff(fact_7837_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z))) ) ).

% distrib_inf_le
tff(fact_7838_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Z))) ) ).

% distrib_sup_le
tff(fact_7839_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P2: A,Q2: A,R: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q2),R))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P2),aa(A,A,uminus_uminus(A),Q2))),R) ) ) ).

% sup_neg_inf
tff(fact_7840_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),aa(A,A,uminus_uminus(A),Y))),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ) ).

% shunt2
tff(fact_7841_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z)) ) ) ).

% shunt1
tff(fact_7842_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B4: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ).

% less_eq_Inf_inter
tff(fact_7843_Un__Inter,axiom,
    ! [A: $tType,A3: set(A),B4: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B4)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3)),B4)) ).

% Un_Inter
tff(fact_7844_SUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M6: fun(B,A),A3: set(B),B4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,M6),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,M6),B4))) ) ).

% SUP_union
tff(fact_7845_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),B4: set(B),A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),B4))),A2) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_awl(fun(B,A),fun(A,fun(B,A)),F2),A2)),B4)) ) ).

% INF_sup
tff(fact_7846_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B4: set(A),A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B4)),A2) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image(A,A,aTP_Lamp_awm(A,fun(A,A),A2)),B4)) ) ).

% Inf_sup
tff(fact_7847_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A2: A,F2: fun(B,A),B4: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),B4))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_awn(A,fun(fun(B,A),fun(B,A)),A2),F2)),B4)) ) ).

% sup_INF
tff(fact_7848_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(C,A),B4: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,G),B4))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_awp(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B4)),A3)) ) ).

% INF_sup_distrib2
tff(fact_7849_SUP__absorb,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [K: A,I5: set(A),A3: fun(A,B)] :
          ( aa(set(A),$o,member(A,K),I5)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,A3,K)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,A3),I5))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,A3),I5)) ) ) ) ).

% SUP_absorb
tff(fact_7850_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,G),A3))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_awq(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A3)) ) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_7851_Un__INT__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B),B4: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),B4),J4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_aws(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A3),B4),J4)),I5)) ).

% Un_INT_distrib2
tff(fact_7852_Un__INT__distrib,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A3: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_awg(set(A),fun(fun(B,set(A)),fun(B,set(A))),B4),A3)),I5)) ).

% Un_INT_distrib
tff(fact_7853_INT__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C5: set(B),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),B4) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_awh(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C5)) ).

% INT_extend_simps(6)
tff(fact_7854_INT__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_awg(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5)) ).

% INT_extend_simps(7)
tff(fact_7855_Un__Union__image,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B4: fun(B,set(A)),C5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_awt(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) ).

% Un_Union_image
tff(fact_7856_UN__Un__distrib,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),B4: fun(B,set(A)),I5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_awt(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),I5))) ).

% UN_Un_distrib
tff(fact_7857_UN__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I5: set(A),A3: fun(A,set(B))] :
      ( aa(set(A),$o,member(A,K),I5)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),A3,K)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)) ) ) ).

% UN_absorb
tff(fact_7858_open__bool__def,axiom,
    topolo1002775350975398744n_open($o) = topolo8378437560675496660pology($o,aa(set(set($o)),set(set($o)),aa(set(set($o)),fun(set(set($o)),set(set($o))),sup_sup(set(set($o))),aa(set($o),set(set($o)),image($o,set($o),set_ord_lessThan($o)),top_top(set($o)))),aa(set($o),set(set($o)),image($o,set($o),set_ord_greaterThan($o)),top_top(set($o))))) ).

% open_bool_def
tff(fact_7859_SUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A2: B,A3: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),aa(set(B),set(B),insert(B,A2),A3))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,A2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F2),A3))) ) ).

% SUP_insert
tff(fact_7860_INF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M6: fun(B,A),A3: set(B),B4: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,M6),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,M6),B4))) ) ).

% INF_union
tff(fact_7861_UN__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B4: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),C5))) = $ite(C5 = bot_bot(set(B)),A3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_awg(set(A),fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C5))) ).

% UN_extend_simps(3)
tff(fact_7862_UN__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),C5: set(B),B4: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),C5))),B4) = $ite(C5 = bot_bot(set(B)),B4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_awh(fun(B,set(A)),fun(set(A),fun(B,set(A))),A3),B4)),C5))) ).

% UN_extend_simps(2)
tff(fact_7863_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),C5: set(B),G: fun(A,B),B4: set(A),D5: set(B)] :
      ( bij_betw(A,B,F2,A3,C5)
     => ( bij_betw(A,B,G,B4,D5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C5),D5) = bot_bot(set(B)) )
           => bij_betw(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_awu(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C5),D5)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_7864_INT__Un,axiom,
    ! [A: $tType,B: $tType,M6: fun(B,set(A)),A3: set(B),B4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),M6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),M6),A3))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),M6),B4))) ).

% INT_Un
tff(fact_7865_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_7866_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B4)) ) ) ) ) ).

% sum.union_inter
tff(fact_7867_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ).

% prod.union_inter
tff(fact_7868_card__Un__Int,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(A,B4)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ).

% card_Un_Int
tff(fact_7869_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_7870_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_7871_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_7872_SUP__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: fun($o,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set($o),set(A),image($o,A,A3),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa($o,A,A3,$true)),aa($o,A,A3,$false)) ) ).

% SUP_UNIV_bool_expand
tff(fact_7873_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or5935395276787703475ssThan(A,Mb,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_7874_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_7875_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_7876_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,Mb)),set_or7035219750837199246ssThan(A,Mb,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_7877_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: A,B4: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_ut(A,fun(nat,A),B4)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B4) ) ).

% SUP_nat_binary
tff(fact_7878_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_7879_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( A3 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image(A,B,F2),A3))
           => ( condit941137186595557371_above(B,aa(set(A),set(B),image(A,B,G),A3))
             => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F2),A3))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G),A3))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_awv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),A3)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_7880_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: set(A),G: fun(A,B),B4: set(A)] :
      ( inj_on(A,B,F2,A3)
     => ( inj_on(A,B,G,B4)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F2),A3)),aa(set(A),set(B),image(A,B,G),B4)) = bot_bot(set(B)) )
         => inj_on(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_awu(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A3),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_7881_Un__eq__UN,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_aww(set(A),fun(set(A),fun($o,set(A))),A3),B4)),top_top(set($o)))) ).

% Un_eq_UN
tff(fact_7882_UN__bool__eq,axiom,
    ! [A: $tType,A3: fun($o,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image($o,set(A),A3),top_top(set($o)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa($o,set(A),A3,$true)),aa($o,set(A),A3,$false)) ).

% UN_bool_eq
tff(fact_7883_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_awx(A,fun(A,$o)),aTP_Lamp_awy(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_7884_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( condit941137186595557371_above(A,A3)
         => ( condit941137186595557371_above(A,B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B4))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_7885_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B4)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7886_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),B4: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ) ).

% sum_Un
tff(fact_7887_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B4)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7888_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7889_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B4)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7890_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ) ).

% sum.union_diff2
tff(fact_7891_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B4: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ).

% sum_Un2
tff(fact_7892_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),insert(A,U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_7893_card__Un__disjoint,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(A,B4)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = bot_bot(set(A)) )
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B4)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_7894_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),B4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ) ).

% prod.union_diff2
tff(fact_7895_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: set(A),B4: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4))
    <=> ( inj_on(A,B,F2,A3)
        & inj_on(A,B,F2,B4)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),B4))),aa(set(A),set(B),image(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B4),A3))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_7896_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_7897_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,Mb)),set_or5935395276787703475ssThan(A,Mb,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_7898_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_7899_sum__Un__nat,axiom,
    ! [A: $tType,A3: set(A),B4: set(A),F2: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(A,B4)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),B4))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_7900_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,Mb: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),Mb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,Mb)),set_or1337092689740270186AtMost(A,Mb,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_7901_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),insert(A,U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_7902_open__nat__def,axiom,
    topolo1002775350975398744n_open(nat) = topolo8378437560675496660pology(nat,aa(set(set(nat)),set(set(nat)),aa(set(set(nat)),fun(set(set(nat)),set(set(nat))),sup_sup(set(set(nat))),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_greaterThan(nat)),top_top(set(nat))))) ).

% open_nat_def
tff(fact_7903_Pow__set_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      pow2(A,aa(list(A),set(A),set2(A),cons(A,Xb,Xs))) = $let(
        a3: set(set(A)),
        a3:= pow2(A,aa(list(A),set(A),set2(A),Xs)),
        aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),a3),aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,Xb)),a3)) ) ).

% Pow_set(2)
tff(fact_7904_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,Xb),A3)
           => ( linord4507533701916653071of_set(A,A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),Xb),linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_7905_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A3: set(A),B4: set(A),F2: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B4)
           => ( ! [X2: A] :
                  ( aa(set(A),$o,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))
                 => ( aa(A,B,F2,X2) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B4))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4))) ) ) ) ) ) ).

% prod_Un
tff(fact_7906_UN__le__eq__Un0,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),aa(nat,set(nat),set_ord_atMost(nat),Nb))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)))),aa(nat,set(A),M6,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_7907_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),union(A,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_union
tff(fact_7908_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C5: set(A),B4: set(A),Xb: A] :
      ( inj_on(A,B,G,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))
       => aa(set(fun(B,A)),$o,member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_awz(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C5),Xb)),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_7909_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A2: A,A3: set(A),B2: B,B4: set(B)] : product_Sigma(A,B,aa(set(A),set(A),insert(A,A2),A3),aa(set(B),fun(A,set(B)),aTP_Lamp_axa(B,fun(set(B),fun(A,set(B))),B2),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A3,aa(set(B),fun(A,set(B)),aTP_Lamp_axa(B,fun(set(B),fun(A,set(B))),B2),B4))),product_Sigma(A,B,aa(set(A),set(A),insert(A,A2),A3),aTP_Lamp_aun(set(B),fun(A,set(B)),B4)))) ).

% insert_Times_insert
tff(fact_7910_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A2: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q)))
     => ( ! [X2: A,Y4: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),X2)),transitive_rtrancl(A,P))
           => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4)),Q)
             => ( X2 = Y4 ) ) )
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,P)) ) ) ).

% rtrancl_Un_separatorE
tff(fact_7911_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A2: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q)))
     => ( ! [X2: A,Y4: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),B2)),transitive_rtrancl(A,P))
           => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X2)),Q)
             => ( Y4 = X2 ) ) )
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),transitive_rtrancl(A,P)) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_7912_sup__set__def,axiom,
    ! [A: $tType,A3: set(A),B4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B4) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B4))) ).

% sup_set_def
tff(fact_7913_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_7914_sup__Un__eq,axiom,
    ! [A: $tType,R2: set(A),S: set(A),X4: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),R2)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),X4)
    <=> aa(set(A),$o,member(A,X4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R2),S)) ) ).

% sup_Un_eq
tff(fact_7915_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B)),X4: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_qm(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_qm(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X4),Xa)
    <=> aa(set(product_prod(A,B)),$o,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)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R2),S)) ) ).

% sup_Un_eq2
tff(fact_7916_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J)
     => ( set_or7035219750837199246ssThan(nat,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I2,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_7917_rtrancl__insert,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_axb(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A2),B2),R)))) ).

% rtrancl_insert
tff(fact_7918_trancl__insert2,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_axc(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A2),B2),R)))) ).

% trancl_insert2
tff(fact_7919_product__atMost__eq__Un,axiom,
    ! [A3: set(nat),Mb: nat] : product_Sigma(nat,nat,A3,aTP_Lamp_axd(nat,fun(nat,set(nat)),Mb)) = aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),sup_sup(set(product_prod(nat,nat))),product_Sigma(nat,nat,A3,aTP_Lamp_avv(nat,fun(nat,set(nat)),Mb))),product_Sigma(nat,nat,A3,aTP_Lamp_axe(nat,fun(nat,set(nat)),Mb))) ).

% product_atMost_eq_Un
tff(fact_7920_trancl__insert,axiom,
    ! [A: $tType,Y: A,Xb: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xb)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_axb(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Y),Xb),R)))) ).

% trancl_insert
tff(fact_7921_Pow__fold,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ( pow2(A,A3) = finite_fold(A,set(set(A)),aTP_Lamp_axf(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),insert(set(A),bot_bot(set(A))),bot_bot(set(set(A)))),A3) ) ) ).

% Pow_fold
tff(fact_7922_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B4: set(A),Xb: B,A3: set(product_prod(B,A))] :
      ( finite_finite2(A,B4)
     => ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(A),set(set(product_prod(B,A))),image(A,set(product_prod(B,A)),aTP_Lamp_axg(B,fun(A,set(product_prod(B,A))),Xb)),B4))),A3) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_axh(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Xb),A3,B4) ) ) ).

% fold_union_pair
tff(fact_7923_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G),zero_zero(A),A3) ) ).

% sum.eq_fold
tff(fact_7924_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,times_times(A)),G),one_one(A),A3) ) ).

% prod.eq_fold
tff(fact_7925_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),set(B),image(A,B,F2),A3) = finite_fold(A,set(B),aTP_Lamp_axi(fun(A,B),fun(A,fun(set(B),set(B))),F2),bot_bot(set(B)),A3) ) ) ).

% image_fold_insert
tff(fact_7926_product__fold,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B4: set(B)] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(B,B4)
       => ( product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_axk(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B4),bot_bot(set(product_prod(A,B))),A3) ) ) ) ).

% product_fold
tff(fact_7927_Id__on__fold,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ( id_on(A,A3) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_axl(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A3) ) ) ).

% Id_on_fold
tff(fact_7928_Set__filter__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( finite_finite2(A,A3)
     => ( filter2(A,P,A3) = finite_fold(A,set(A),aTP_Lamp_axm(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A3) ) ) ).

% Set_filter_fold
tff(fact_7929_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),Xb: product_prod(C,A),X5: set(product_prod(C,B))] :
      ( finite_finite2(product_prod(A,B),S)
     => ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert(product_prod(C,A),Xb),bot_bot(set(product_prod(C,A)))),S)),X5) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_axn(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xb)),X5,S) ) ) ).

% insert_relcomp_union_fold
tff(fact_7930_relpow__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),R2) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Mb),R2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2)) ).

% relpow_add
tff(fact_7931_relpow_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,Nb)),R2) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R2),R2) ).

% relpow.simps(2)
tff(fact_7932_relcomp_Ocases,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R: set(product_prod(A,C)),S3: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22)),relcomp(A,C,B,R,S3))
     => ~ ! [B3: C] :
            ( aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),B3)),R)
           => ~ aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B3),A22)),S3) ) ) ).

% relcomp.cases
tff(fact_7933_relcomp_Osimps,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R: set(product_prod(A,C)),S3: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),A22)),relcomp(A,C,B,R,S3))
    <=> ? [A5: A,B5: C,C4: B] :
          ( ( A1 = A5 )
          & ( A22 = C4 )
          & aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A5),B5)),R)
          & aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B5),C4)),S3) ) ) ).

% relcomp.simps
tff(fact_7934_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R: set(product_prod(A,B)),C2: C,S3: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,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)),R)
     => ( aa(set(product_prod(B,C)),$o,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C2)),S3)
       => aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A2),C2)),relcomp(A,B,C,R,S3)) ) ) ).

% relcomp.relcompI
tff(fact_7935_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R: set(product_prod(A,C)),S3: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),Xz),relcomp(A,C,B,R,S3))
     => ~ ! [X2: A,Y4: C,Z4: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Z4) )
           => ( aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X2),Y4)),R)
             => ~ aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y4),Z4)),S3) ) ) ) ).

% relcompE
tff(fact_7936_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A2: A,C2: B,R: set(product_prod(A,C)),S3: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),C2)),relcomp(A,C,B,R,S3))
     => ~ ! [B3: C] :
            ( aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A2),B3)),R)
           => ~ aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B3),C2)),S3) ) ) ).

% relcompEpair
tff(fact_7937_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,$o),A3: set(A)] : filter2(A,P,A3) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_axo(fun(A,$o),fun(set(A),fun(A,$o)),P),A3)) ).

% Set.filter_def
tff(fact_7938_relcomp__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,B)),A3: set(A),B4: set(B),S3: set(product_prod(B,C)),C5: set(C)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),product_Sigma(A,B,A3,aTP_Lamp_aun(set(B),fun(A,set(B)),B4)))
     => ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),S3),product_Sigma(B,C,B4,aTP_Lamp_avk(set(C),fun(B,set(C)),C5)))
       => aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R,S3)),product_Sigma(A,C,A3,aTP_Lamp_axp(set(C),fun(A,set(C)),C5))) ) ) ).

% relcomp_subset_Sigma
tff(fact_7939_relcomp__UNION__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S3: set(product_prod(A,C)),R: fun(D,set(product_prod(C,B))),I5: set(D)] : relcomp(A,C,B,S3,aa(set(set(product_prod(C,B))),set(product_prod(C,B)),complete_Sup_Sup(set(product_prod(C,B))),aa(set(D),set(set(product_prod(C,B))),image(D,set(product_prod(C,B)),R),I5))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_axq(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),S3),R)),I5)) ).

% relcomp_UNION_distrib
tff(fact_7940_relcomp__UNION__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R: fun(D,set(product_prod(A,C))),I5: set(D),S3: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(set(product_prod(A,C))),set(product_prod(A,C)),complete_Sup_Sup(set(product_prod(A,C))),aa(set(D),set(set(product_prod(A,C))),image(D,set(product_prod(A,C)),R),I5)),S3) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_axr(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),R),S3)),I5)) ).

% relcomp_UNION_distrib2
tff(fact_7941_card_Oeq__fold,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = finite_fold(A,nat,aTP_Lamp_axs(A,fun(nat,nat)),zero_zero(nat),A3) ).

% card.eq_fold
tff(fact_7942_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : linord4507533701916653071of_set(A,A3) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_asy(A,A)),nil(A),A3) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_7943_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(B,C))] :
      ( finite_finite2(product_prod(A,B),R2)
     => ( finite_finite2(product_prod(B,C),S)
       => ( relcomp(A,B,C,R2,S) = finite_fold(product_prod(A,B),set(product_prod(A,C)),aa(fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(A,B),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_axu(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S)),bot_bot(set(product_prod(A,C))),R2) ) ) ) ).

% relcomp_fold
tff(fact_7944_inter__Set__filter,axiom,
    ! [A: $tType,B4: set(A),A3: set(A)] :
      ( finite_finite2(A,B4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B4) = filter2(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A3),B4) ) ) ).

% inter_Set_filter
tff(fact_7945_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),Xb: product_prod(C,A),R2: set(product_prod(C,A))] :
      ( finite_finite2(product_prod(A,B),S)
     => ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert(product_prod(C,A),Xb),R2),S) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_axn(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xb)),relcomp(C,A,B,R2,S),S) ) ) ).

% insert_relcomp_fold
tff(fact_7946_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( finite_finite2(product_prod(A,B),S)
     => finite6289374366891150609ommute(product_prod(C,A),set(product_prod(C,B)),aa(fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(C,A),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_axw(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_7947_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_axx(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Max.eq_fold'
tff(fact_7948_option_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),Option: option(B)] :
      case_option(A,B,F1,F22,Option) = $ite(Option = none(B),F1,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ).

% option.case_eq_if
tff(fact_7949_option_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,B),Option: option(C)] : aa(B,A,H,case_option(B,C,F1,F22,Option)) = case_option(A,C,aa(B,A,H,F1),aa(fun(C,B),fun(C,A),aTP_Lamp_mr(fun(B,A),fun(fun(C,B),fun(C,A)),H),F22),Option) ).

% option.case_distrib
tff(fact_7950_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_axy(fun(B,B),fun(A,fun(B,B)),F2)) ).

% comp_fun_commute_const
tff(fact_7951_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_tn(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_7952_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_ag(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_7953_option_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,A)] : case_option(A,B,F1,F22,none(B)) = F1 ).

% option.simps(4)
tff(fact_7954_option_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),X23: B] : case_option(A,B,F1,F22,aa(B,option(B),some(B),X23)) = aa(B,A,F22,X23) ).

% option.simps(5)
tff(fact_7955_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,$o),Xb: option(A),R2: fun(B,$o),F2: B,G: fun(A,B)] :
      ( case_option($o,A,(P),Q,Xb)
     => ( ( (P)
         => aa(B,$o,R2,F2) )
       => ( ! [Q3: A] :
              ( aa(A,$o,Q,Q3)
             => aa(B,$o,R2,aa(A,B,G,Q3)) )
         => aa(B,$o,R2,case_option(B,A,F2,G,Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_7956_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),Xb: option(A)] :
      ( case_option($o,A,(P),Q,Xb)
     => ( ( ( Xb = none(A) )
         => ~ (P) )
       => ~ ! [Y4: A] :
              ( ( Xb = aa(A,option(A),some(A),Y4) )
             => ~ aa(A,$o,Q,Y4) ) ) ) ).

% case_optionE
tff(fact_7957_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P: fun(A,$o)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_axm(fun(A,$o),fun(A,fun(set(A),set(A))),P)) ).

% comp_fun_commute_filter_fold
tff(fact_7958_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite6289374366891150609ommute(A,B,F2)
     => finite6289374366891150609ommute(A,B,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_axz(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F2),G)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_7959_comp__fun__commute__Image__fold,axiom,
    ! [B: $tType,A: $tType,S: set(A)] : finite6289374366891150609ommute(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_aya(set(A),fun(A,fun(B,fun(set(B),set(B)))),S))) ).

% comp_fun_commute_Image_fold
tff(fact_7960_option_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F1,F22,Option))
    <=> ( ( ( Option = none(B) )
         => aa(A,$o,P,F1) )
        & ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
         => aa(A,$o,P,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel
tff(fact_7961_option_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F1,F22,Option))
    <=> ~ ( ( ( Option = none(B) )
            & ~ aa(A,$o,P,F1) )
          | ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
            & ~ aa(A,$o,P,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_7962_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B4: set(A)] :
      ( finite_finite2(A,B4)
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_ayb(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B4)) ) ).

% comp_fun_commute_product_fold
tff(fact_7963_take__bit__numeral__minus__numeral__int,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = case_option(int,num,zero_zero(int),aTP_Lamp_ayc(num,fun(num,int),Mb),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),Mb),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_7964_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: 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,bit0,Nb)))),aa(num,int,numeral_numeral(int),Mb)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Mb,bitM(Nb))) ).

% and_minus_numerals(7)
tff(fact_7965_take__bit__num__simps_I1_J,axiom,
    ! [Mb: num] : bit_take_bit_num(zero_zero(nat),Mb) = none(num) ).

% take_bit_num_simps(1)
tff(fact_7966_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa(nat,nat,suc,Nb),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(2)
tff(fact_7967_take__bit__num__simps_I5_J,axiom,
    ! [R: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_7968_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa(nat,nat,suc,Nb),aa(num,num,bit0,Mb)) = case_option(option(num),num,none(num),aTP_Lamp_ayd(num,option(num)),bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_7969_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa(nat,nat,suc,Nb),aa(num,num,bit1,Mb)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_7970_take__bit__num__simps_I6_J,axiom,
    ! [R: num,Mb: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit0,Mb)) = case_option(option(num),num,none(num),aTP_Lamp_ayd(num,option(num)),bit_take_bit_num(pred_numeral(R),Mb)) ).

% take_bit_num_simps(6)
tff(fact_7971_take__bit__num__simps_I7_J,axiom,
    ! [R: num,Mb: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit1,Mb)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R),Mb))) ).

% take_bit_num_simps(7)
tff(fact_7972_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: num,Nb: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),Mb)),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),Mb),Nb)) ) ).

% take_bit_numeral_numeral
tff(fact_7973_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: 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,Nb)))),aa(num,int,numeral_numeral(int),Mb)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Mb,aa(num,num,bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_7974_and__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Mb,aa(num,num,bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_7975_and__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,Nb)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Mb,bitM(Nb))) ).

% and_minus_numerals(3)
tff(fact_7976_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa(num,num,bit0,Mb)) = case_nat(option(num),none(num),aTP_Lamp_aye(num,fun(nat,option(num)),Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_7977_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa(num,num,bit1,Mb)) = case_nat(option(num),none(num),aTP_Lamp_ayf(num,fun(nat,option(num)),Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_7978_comp__fun__commute__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_asy(A,A))) ) ).

% comp_fun_commute_insort
tff(fact_7979_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: num,Q2: num] :
          ( ( bit_take_bit_num(Mb,Nb) = aa(num,option(num),some(num),Q2) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_7980_and__not__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa(num,num,bit0,Mb),one2) = aa(num,option(num),some(num),aa(num,num,bit0,Mb)) ).

% and_not_num.simps(4)
tff(fact_7981_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one2,aa(num,num,bit0,Nb)) = aa(num,option(num),some(num),one2) ).

% and_not_num.simps(2)
tff(fact_7982_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one2,aa(num,num,bit1,Nb)) = none(num) ).

% and_not_num.simps(3)
tff(fact_7983_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_7984_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one2) = case_nat(option(num),none(num),aTP_Lamp_ayg(nat,option(num)),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_7985_and__not__num_Osimps_I7_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa(num,num,bit1,Mb),one2) = aa(num,option(num),some(num),aa(num,num,bit0,Mb)) ).

% and_not_num.simps(7)
tff(fact_7986_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: num] :
          ( ( bit_take_bit_num(Mb,Nb) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_7987_and__not__num__eq__Some__iff,axiom,
    ! [Mb: num,Nb: num,Q2: num] :
      ( ( bit_and_not_num(Mb,Nb) = 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),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))) = aa(num,int,numeral_numeral(int),Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_7988_and__not__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa(num,num,bit1,Mb),aa(num,num,bit0,Nb)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_ayh(num,option(num)),bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(8)
tff(fact_7989_and__not__num__eq__None__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( bit_and_not_num(Mb,Nb) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_7990_int__numeral__not__and__num,axiom,
    ! [Mb: num,Nb: 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),Mb))),aa(num,int,numeral_numeral(int),Nb)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Nb,Mb)) ).

% int_numeral_not_and_num
tff(fact_7991_int__numeral__and__not__num,axiom,
    ! [Mb: num,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),Mb)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Nb))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Mb,Nb)) ).

% int_numeral_and_not_num
tff(fact_7992_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(num,nat,numeral_numeral(nat),Mb)) = zero_zero(nat),none(num),aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(num,nat,numeral_numeral(nat),Mb))))) ).

% take_bit_num_def
tff(fact_7993_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_axf(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_7994_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                    $let(
                      m2: nat,
                      m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                      ( ! [X: vEBT_VEBT] :
                          ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_VEBT_valid(X,n) )
                      & vEBT_VEBT_valid(Summary,m2)
                      & ( 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),aa(num,num,bit0,one2))),m2) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                          & ! [X: vEBT_VEBT] :
                              ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                          aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ayi(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_7995_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                      $let(
                        m2: nat,
                        m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                        ( ! [X: vEBT_VEBT] :
                            ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_VEBT_valid(X,n) )
                        & vEBT_VEBT_valid(Summary,m2)
                        & ( 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),aa(num,num,bit0,one2))),m2) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                            & ! [X: vEBT_VEBT] :
                                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                            aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ayi(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_7996_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,fun(A,$o))] :
      ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),P))
     => ( finite_finite2(B,aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_ayj(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),P),Q)))
      <=> ! [Y3: A] :
            ( aa(A,$o,P,Y3)
           => finite_finite2(B,aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_ayk(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),Y3))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_7997_INF__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_ayl(set(A),fun(fun(A,$o),$o)) = ball(A) ).

% INF_bool_eq
tff(fact_7998_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: set(product_prod(A,C)),S3: set(product_prod(C,B))] : relcomp(A,C,B,R,S3) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_aym(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),R),S3))) ).

% relcomp_unfold
tff(fact_7999_Collect__ex__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ayn(fun(A,fun(B,$o)),fun(A,$o),P)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_anw(fun(A,fun(B,$o)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_ex_eq
tff(fact_8000_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ayo(fun(B,A),fun(A,$o),F2)) = aa(set(B),set(A),image(B,A,F2),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_8001_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_ayp(fun(B,A),fun(fun(B,$o),fun(A,$o)),F2),P)) = aa(set(B),set(A),image(B,A,F2),aa(fun(B,$o),set(B),collect(B),P)) ).

% setcompr_eq_image
tff(fact_8002_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: set(B)] : aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ayq(fun(B,A),fun(set(B),fun(A,$o)),F2),A3)) = aa(set(B),set(A),image(B,A,F2),A3) ).

% Setcompr_eq_image
tff(fact_8003_Collect__ball__eq,axiom,
    ! [A: $tType,B: $tType,A3: set(B),P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,fun(B,$o)),fun(A,$o),aTP_Lamp_ayr(set(B),fun(fun(A,fun(B,$o)),fun(A,$o)),A3),P)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_anw(fun(A,fun(B,$o)),fun(B,set(A)),P)),A3)) ).

% Collect_ball_eq
tff(fact_8004_INTER__eq,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A3: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B4),A3)) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ays(fun(B,set(A)),fun(set(B),fun(A,$o)),B4),A3)) ).

% INTER_eq
tff(fact_8005_open__Collect__ex,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,fun(B,$o))] :
          ( ! [I3: A] : aa(set(B),$o,topolo1002775350975398744n_open(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),P,I3)))
         => aa(set(B),$o,topolo1002775350975398744n_open(B),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_ayt(fun(A,fun(B,$o)),fun(B,$o),P))) ) ) ).

% open_Collect_ex
tff(fact_8006_open__diagonal__complement,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => aa(set(product_prod(A,A)),$o,topolo1002775350975398744n_open(product_prod(A,A)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_ayu(product_prod(A,A),$o))) ) ).

% open_diagonal_complement
tff(fact_8007_Ball__comp__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: fun(B,set(C)),F2: fun(C,$o),G: fun(A,B),X4: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),comp(B,$o,A,aa(fun(C,$o),fun(B,$o),aTP_Lamp_ayv(fun(B,set(C)),fun(fun(C,$o),fun(B,$o)),A3),F2)),G),X4)
    <=> ! [Xa2: C] :
          ( aa(set(C),$o,member(C,Xa2),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),comp(B,set(C),A,A3),G),X4))
         => aa(C,$o,F2,Xa2) ) ) ).

% Ball_comp_iff
tff(fact_8008_set__Cons__def,axiom,
    ! [A: $tType,A3: set(A),XS: set(list(A))] : set_Cons(A,A3,XS) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(list(A)),fun(list(A),$o),aTP_Lamp_ayw(set(A),fun(set(list(A)),fun(list(A),$o)),A3),XS)) ).

% set_Cons_def
tff(fact_8009_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_ayx(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ).

% lexordp.mono
tff(fact_8010_ord_Olexordp_Omono,axiom,
    ! [A: $tType,Less: fun(A,fun(A,$o))] : order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_ayy(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Less)) ).

% ord.lexordp.mono
tff(fact_8011_rtranclp_Omono,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] : order_mono(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_ayz(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),R)) ).

% rtranclp.mono
tff(fact_8012_tranclp_Omono,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] : order_mono(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_aza(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),R)) ).

% tranclp.mono
tff(fact_8013_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_ayn(fun(A,fun(B,$o)),fun(A,$o),P),F4)
    <=> ? [Y8: fun(A,B)] : eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_azb(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),P),Y8),F4) ) ).

% eventually_ex
tff(fact_8014_Union__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),P: fun(B,$o)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(fun(B,$o),fun(set(A),$o),aTP_Lamp_azc(fun(B,set(A)),fun(fun(B,$o),fun(set(A),$o)),F2),P))) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_azd(fun(B,set(A)),fun(fun(B,$o),fun(A,$o)),F2),P)) ).

% Union_SetCompr_eq
tff(fact_8015_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aze(set(A),fun(A,$o),A3))) ) ).

% Inf_eq_Sup
tff(fact_8016_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_azf(set(A),fun(A,$o),A3))) ) ).

% Sup_eq_Inf
tff(fact_8017_closed__diagonal,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_azg(product_prod(A,A),$o))) ) ).

% closed_diagonal
tff(fact_8018_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F2: fun(A,B)] :
      ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),P))
     => finite_finite2(B,aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aTP_Lamp_azh(fun(A,$o),fun(fun(A,B),fun(B,$o)),P),F2))) ) ).

% finite_image_set
tff(fact_8019_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,$o),F2: fun(A,fun(B,C))] :
      ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),P))
     => ( finite_finite2(B,aa(fun(B,$o),set(B),collect(B),Q))
       => finite_finite2(C,aa(fun(C,$o),set(C),collect(C),aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_azi(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),P),Q),F2))) ) ) ).

% finite_image_set2
tff(fact_8020_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( finite_finite2(A,A3)
     => ( eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_azj(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),P),Net)
      <=> ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_ayk(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_8021_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A3: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( finite_finite2(A,A3)
     => ( ! [X2: A] :
            ( aa(set(A),$o,member(A,X2),A3)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_ayk(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X2),Net) )
       => eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_azj(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),P),Net) ) ) ).

% eventually_ball_finite
tff(fact_8022_finite_Omono,axiom,
    ! [A: $tType] : order_mono(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_azk(fun(set(A),$o),fun(set(A),$o))) ).

% finite.mono
tff(fact_8023_Gr__def,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F2: fun(A,B)] : bNF_Gr(A,B,A3,F2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),$o),aTP_Lamp_azl(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),A3),F2)) ).

% Gr_def
tff(fact_8024_Ball__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( finite_finite2(A,A3)
     => ( ! [X: A] :
            ( aa(set(A),$o,member(A,X),A3)
           => aa(A,$o,P,X) )
      <=> finite_fold(A,$o,aTP_Lamp_azm(fun(A,$o),fun(A,fun($o,$o)),P),$true,A3) ) ) ).

% Ball_fold
tff(fact_8025_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => aa(set(product_prod(A,A)),$o,topolo1002775350975398744n_open(product_prod(A,A)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_azn(product_prod(A,A),$o))) ) ).

% open_superdiagonal
tff(fact_8026_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => aa(set(product_prod(A,A)),$o,topolo1002775350975398744n_open(product_prod(A,A)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_azo(product_prod(A,A),$o))) ) ).

% open_subdiagonal
tff(fact_8027_closed__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_azp(product_prod(A,A),$o))) ) ).

% closed_subdiagonal
tff(fact_8028_closed__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_azq(product_prod(A,A),$o))) ) ).

% closed_superdiagonal
tff(fact_8029_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_azr(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_8030_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A)] :
          ( ( S != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S)
           => ( aa(set(A),A,complete_Inf_Inf(A),S) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_azs(set(A),fun(A,$o),S))) ) ) ) ) ).

% cInf_cSup
tff(fact_8031_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A)] :
          ( ( S != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S)
           => ( aa(set(A),A,complete_Sup_Sup(A),S) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_azt(set(A),fun(A,$o),S))) ) ) ) ) ).

% cSup_cInf
tff(fact_8032_funpow__inj__finite,axiom,
    ! [A: $tType,P2: fun(A,A),Xb: A] :
      ( inj_on(A,A,P2,top_top(set(A)))
     => ( finite_finite2(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_azu(fun(A,A),fun(A,fun(A,$o)),P2),Xb)))
       => ~ ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),P2),Xb) != Xb ) ) ) ) ).

% funpow_inj_finite
tff(fact_8033_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Dega: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Deg3: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Dega,TreeLista,Summarya),Deg3)
    <=> ( ( Dega = Deg3 )
        & $let(
            n: nat,
            n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Dega),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
            $let(
              m2: nat,
              m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Dega),n),
              ( ! [X: vEBT_VEBT] :
                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                 => vEBT_VEBT_valid(X,n) )
              & vEBT_VEBT_valid(Summarya,m2)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X7)
                  & ! [X: vEBT_VEBT] :
                      ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                  aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ayi(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Dega),TreeLista),Summarya),n),m2)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_8034_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( 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),aa(num,num,bit0,one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X: vEBT_VEBT] :
                                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                              aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ayi(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_8035_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (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),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = 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))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( 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),aa(num,num,bit0,one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X: vEBT_VEBT] :
                                    ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ayi(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),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,Deg,TreeList,Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_8036_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( 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),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = 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))),Xaa))
               => ( Xaa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( 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,Deg,TreeList,Summary)),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( 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),aa(num,num,bit0,one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X: vEBT_VEBT] :
                                    ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ayi(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_8037_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_azv(set(set(A)),fun(set(A),$o),A3))))) ) ).

% Inf_Sup_le
tff(fact_8038_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_azw(set(set(A)),fun(set(A),$o),A3))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3))) ) ).

% Sup_Inf_le
tff(fact_8039_SUP__INF__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: fun(B,A),A3: set(set(B))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),image(set(B),A,aTP_Lamp_azx(fun(B,A),fun(set(B),A),G)),A3)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),image(set(B),A,aTP_Lamp_azy(fun(B,A),fun(set(B),A),G)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_azz(set(set(B)),fun(set(B),$o),A3)))) ) ).

% SUP_INF_set
tff(fact_8040_INF__SUP__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: fun(B,A),A3: set(set(B))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),image(set(B),A,aTP_Lamp_azy(fun(B,A),fun(set(B),A),G)),A3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),image(set(B),A,aTP_Lamp_azx(fun(B,A),fun(set(B),A),G)),aa(fun(set(B),$o),set(set(B)),collect(set(B)),aTP_Lamp_azz(set(set(B)),fun(set(B),$o),A3)))) ) ).

% INF_SUP_set
tff(fact_8041_Inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: set(set(A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_azv(set(set(A)),fun(set(A),$o),A3)))) ) ).

% Inf_Sup
tff(fact_8042_Sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: set(set(A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),A3)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_azv(set(set(A)),fun(set(A),$o),A3)))) ) ).

% Sup_Inf
tff(fact_8043_Inter__eq,axiom,
    ! [A: $tType,A3: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_baa(set(set(A)),fun(A,$o),A3)) ).

% Inter_eq
tff(fact_8044_Sup__int__def,axiom,
    ! [X5: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X5) = the(int,aTP_Lamp_bab(set(int),fun(int,$o),X5)) ).

% Sup_int_def
tff(fact_8045_Pow__Compl,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A3)) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_bac(set(A),fun(set(A),$o),A3)) ).

% Pow_Compl
tff(fact_8046_mono__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Q: fun(A,fun(B,C)),F2: fun(D,B)] :
          ( order_mono(A,fun(B,C),Q)
         => order_mono(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_bad(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F2)) ) ) ).

% mono_compose
tff(fact_8047_Union__maximal__sets,axiom,
    ! [A: $tType,F8: set(set(A))] :
      ( finite_finite2(set(A),F8)
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_bae(set(set(A)),fun(set(A),$o),F8))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F8) ) ) ).

% Union_maximal_sets
tff(fact_8048_Inf__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),S) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(fun(filter(A),$o),set(filter(A)),collect(filter(A)),aTP_Lamp_baf(set(filter(A)),fun(filter(A),$o),S))) ).

% Inf_filter_def
tff(fact_8049_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( 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),Xb),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xb = 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))),Xaa))
               => ( Xaa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( 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,Deg,TreeList,Summary)),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( 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),aa(num,num,bit0,one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X: vEBT_VEBT] :
                                  ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                              aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ayi(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_8050_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,Mb) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_baj(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),Nb),Mb)) ).

% Bit_Operations.take_bit_num_code
tff(fact_8051_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A3: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_bak(set(set(A)),fun(set(A),$o),A3))))) ) ).

% finite_Inf_Sup
tff(fact_8052_verit__eq__simplify_I18_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F33: fun(num,A),X32: num] : case_num(A,F1,F22,F33,aa(num,num,bit1,X32)) = aa(num,A,F33,X32) ).

% verit_eq_simplify(18)
tff(fact_8053_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F33: fun(num,A),X23: num] : case_num(A,F1,F22,F33,aa(num,num,bit0,X23)) = aa(num,A,F22,X23) ).

% verit_eq_simplify(17)
tff(fact_8054_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F33: fun(num,A)] : case_num(A,F1,F22,F33,one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_8055_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(num,B),F33: fun(num,B),Num: num] : aa(B,A,H,case_num(B,F1,F22,F33,Num)) = case_num(A,aa(B,A,H,F1),aa(fun(num,B),fun(num,A),aTP_Lamp_bal(fun(B,A),fun(fun(num,B),fun(num,A)),H),F22),aa(fun(num,B),fun(num,A),aTP_Lamp_bal(fun(B,A),fun(fun(num,B),fun(num,A)),H),F33),Num) ).

% num.case_distrib
tff(fact_8056_finite__inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A2: A,A3: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Sup_Sup(A),A3)) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_bam(A,fun(set(A),fun(A,$o)),A2),A3))) ) ).

% finite_inf_Sup
tff(fact_8057_lexn__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Nb: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),Nb) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_ban(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R),Nb))) ).

% lexn_conv
tff(fact_8058_mlex__eq,axiom,
    ! [A: $tType,F2: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F2,R2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_bao(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F2),R2))) ).

% mlex_eq
tff(fact_8059_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),Nb: nat] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),Nb))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = Nb ) ) ) ).

% lexn_length
tff(fact_8060_mlex__leq,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
     => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),R2)
       => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),mlex_prod(A,F2,R2)) ) ) ).

% mlex_leq
tff(fact_8061_mlex__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xb: A,Y: A,R2: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
     => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),mlex_prod(A,F2,R2)) ) ).

% mlex_less
tff(fact_8062_mlex__iff,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),mlex_prod(A,F2,R2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y))
        | ( ( aa(A,nat,F2,Xb) = aa(A,nat,F2,Y) )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),R2) ) ) ) ).

% mlex_iff
tff(fact_8063_in__measure,axiom,
    ! [A: $tType,Xb: A,Y: A,F2: fun(A,nat)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),measure(A,F2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xb)),aa(A,nat,F2,Y)) ) ).

% in_measure
tff(fact_8064_lex__prod__def,axiom,
    ! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_baq(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_8065_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B,R: set(product_prod(A,A)),S3: set(product_prod(B,B))] :
      ( aa(set(product_prod(product_prod(A,B),product_prod(A,B))),$o,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6))),lex_prod(A,B,R,S3))
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A6)),R)
        | ( ( A2 = A6 )
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B6)),S3) ) ) ) ).

% in_lex_prod
tff(fact_8066_lex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_bar(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R))) ).

% lex_conv
tff(fact_8067_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] : lattic5882676163264333800up_fin(A,A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_bas(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Sup_fin.eq_fold'
tff(fact_8068_Cons__in__lex,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),cons(A,Xb,Xs)),cons(A,Y,Ys))),lex(A,R))
    <=> ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),R)
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( Xb = Y )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R)) ) ) ) ).

% Cons_in_lex
tff(fact_8069_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),lattic5882676163264333800up_fin(A,A3)) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_8070_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),Xb)
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xb) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_8071_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xb) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),Xb) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_8072_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),Xb)
             => ! [A8: A] :
                  ( aa(set(A),$o,member(A,A8),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A8),Xb) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_8073_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] :
          ( ~ finite_finite2(A,A3)
         => ( lattic5882676163264333800up_fin(A,A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_8074_lex__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = aa(set(set(product_prod(list(A),list(A)))),set(product_prod(list(A),list(A))),complete_Sup_Sup(set(product_prod(list(A),list(A)))),aa(set(nat),set(set(product_prod(list(A),list(A)))),image(nat,set(product_prod(list(A),list(A))),lexn(A,R)),top_top(set(nat)))) ).

% lex_def
tff(fact_8075_lex__append__left__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs2: list(A)] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),R)
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs2))),lex(A,R))
      <=> aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2)),lex(A,R)) ) ) ).

% lex_append_left_iff
tff(fact_8076_lex__append__leftD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs2: list(A)] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),R)
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs2))),lex(A,R))
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2)),lex(A,R)) ) ) ).

% lex_append_leftD
tff(fact_8077_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Us)),append(A,Ys,Vs))),lex(A,R)) ) ) ).

% lex_append_rightI
tff(fact_8078_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A3)),lattic5882676163264333800up_fin(A,B4)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_8079_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),lattic5882676163264333800up_fin(A,A3)) = lattic5882676163264333800up_fin(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_bat(set(A),fun(A,fun(A,$o)),A3),Xb))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_8080_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B4)
             => ( ( B4 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic5882676163264333800up_fin(A,A3)),lattic5882676163264333800up_fin(A,B4)) = lattic5882676163264333800up_fin(A,aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_bau(set(A),fun(set(A),fun(A,$o)),A3),B4))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_8081_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),insert(A,Xb),A3)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_8082_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,Xb),A3)
           => ( lattic5882676163264333800up_fin(A,A3) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_8083_lenlex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_bav(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R))) ).

% lenlex_conv
tff(fact_8084_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] : lattic7752659483105999362nf_fin(A,A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_baw(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Inf_fin.eq_fold'
tff(fact_8085_lenlex__irreflexive,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),R)
     => ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lenlex(A,R)) ) ).

% lenlex_irreflexive
tff(fact_8086_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic7752659483105999362nf_fin(A,A3))
             => ! [A8: A] :
                  ( aa(set(A),$o,member(A,A8),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A8) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_8087_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( aa(set(A),$o,member(A,A4),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic7752659483105999362nf_fin(A,A3)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_8088_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic7752659483105999362nf_fin(A,A3))
            <=> ! [X: A] :
                  ( aa(set(A),$o,member(A,X),A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_8089_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,A2),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A3)),A2) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_8090_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] :
          ( ~ finite_finite2(A,A3)
         => ( lattic7752659483105999362nf_fin(A,A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_8091_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B4)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,B4)),lattic7752659483105999362nf_fin(A,A3)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_8092_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)) ) ).

% lenlex_length
tff(fact_8093_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R2: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xs)),lenlex(A,R2))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Us,Vs)),append(A,Xs,Ys))),lenlex(A,R2)) ) ) ).

% lenlex_append1
tff(fact_8094_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A3)),lattic5882676163264333800up_fin(A,A3)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_8095_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),B4: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B4)
             => ( ( B4 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic7752659483105999362nf_fin(A,A3)),lattic7752659483105999362nf_fin(A,B4)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_bax(set(A),fun(set(A),fun(A,$o)),A3),B4))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_8096_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),lattic7752659483105999362nf_fin(A,A3)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_bay(set(A),fun(A,fun(A,$o)),A3),Xb))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_8097_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,member(A,Xb),A3)
           => ( lattic7752659483105999362nf_fin(A,A3) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_8098_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert(A,Xb),A3)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_8099_Cons__lenlex__iff,axiom,
    ! [A: $tType,Mb: A,Ms: list(A),Nb: A,Ns: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),cons(A,Mb,Ms)),cons(A,Nb,Ns))),lenlex(A,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Mb),Nb)),R) )
        | ( ( Mb = Nb )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_8100_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs2: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs2))) )
    <=> ( ( Xs = append(A,Ys,cons(A,Y,Zs2)) )
        & aa(A,$o,P,Y)
        & ~ ? [X: A] :
              ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X) ) ) ) ).

% extract_Some_iff
tff(fact_8101_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs2: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs2))) )
     => ( ( Xs = append(A,Ys,cons(A,Y,Zs2)) )
        & aa(A,$o,P,Y)
        & ~ ? [X4: A] :
              ( aa(set(A),$o,member(A,X4),aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X4) ) ) ) ).

% extract_SomeE
tff(fact_8102_extract__None__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( extract(A,P,Xs) = none(product_prod(list(A),product_prod(A,list(A)))) )
    <=> ~ ? [X: A] :
            ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X) ) ) ).

% extract_None_iff
tff(fact_8103_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      extract(A,P,cons(A,Xb,Xs)) = $ite(aa(A,$o,P,Xb),aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),nil(A)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Xb),Xs))),case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A))),none(product_prod(list(A),product_prod(A,list(A)))),aa(fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_bba(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Xb)),extract(A,P,Xs))) ).

% extract_Cons_code
tff(fact_8104_lexord__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lexord(A,R) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_bbb(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R))) ).

% lexord_def
tff(fact_8105_listrel1__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : listrel1(A,R) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_bbc(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R))) ).

% listrel1_def
tff(fact_8106_Cons__listrel1__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),cons(A,Xb,Xs)),cons(A,Y,Ys))),listrel1(A,R))
    <=> ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),R)
          & ( Xs = Ys ) )
        | ( ( Xb = Y )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_8107_lexord__cons__cons,axiom,
    ! [A: $tType,A2: A,Xb: list(A),B2: A,Y: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),cons(A,A2,Xb)),cons(A,B2,Y))),lexord(A,R))
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R)
        | ( ( A2 = B2 )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Y)),lexord(A,R)) ) ) ) ).

% lexord_cons_cons
tff(fact_8108_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Y)),transitive_rtrancl(list(A),listrel1(A,R)))
     => ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_8109_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),cons(A,Y,Ys))),listrel1(A,R))
     => ( ! [X2: A] :
            ( ( Xs = cons(A,X2,Ys) )
           => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),R) )
       => ~ ! [Zs: list(A)] :
              ( ( Xs = cons(A,Y,Zs) )
             => ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs),Ys)),listrel1(A,R)) ) ) ) ).

% Cons_listrel1E2
tff(fact_8110_Cons__listrel1E1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),cons(A,Xb,Xs)),Ys)),listrel1(A,R))
     => ( ! [Y4: A] :
            ( ( Ys = cons(A,Y4,Xs) )
           => ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y4)),R) )
       => ~ ! [Zs: list(A)] :
              ( ( Ys = cons(A,Xb,Zs) )
             => ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs)),listrel1(A,R)) ) ) ) ).

% Cons_listrel1E1
tff(fact_8111_listrel1I1,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),Xs: list(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),R)
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),cons(A,Xb,Xs)),cons(A,Y,Xs))),listrel1(A,R)) ) ).

% listrel1I1
tff(fact_8112_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_8113_lexord__irreflexive,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X2: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),R)
     => ~ aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lexord(A,R)) ) ).

% lexord_irreflexive
tff(fact_8114_lexord__linear,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xb: list(A),Y: list(A)] :
      ( ! [A4: A,B3: A] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3)),R)
          | ( A4 = B3 )
          | aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4)),R) )
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Y)),lexord(A,R))
        | ( Xb = Y )
        | aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Xb)),lexord(A,R)) ) ) ).

% lexord_linear
tff(fact_8115_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),Ys: list(A),Zs2: list(A)] :
      ( ! [X2: A,Y4: A,Z4: A] :
          ( aa(set(A),$o,member(A,X2),aa(list(A),set(A),set2(A),Xs))
         => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4)),R)
           => ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R)
             => aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z4)),R) ) ) )
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R))
       => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2)),lexord(A,R))
         => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2)),lexord(A,R)) ) ) ) ).

% lexord_partial_trans
tff(fact_8116_lexord__append__leftD,axiom,
    ! [A: $tType,Xb: list(A),U: list(A),V: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xb,U)),append(A,Xb,V))),lexord(A,R))
     => ( ! [A4: A] : ~ aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4)),R)
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V)),lexord(A,R)) ) ) ).

% lexord_append_leftD
tff(fact_8117_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs2: list(A),Ys: list(A),Qs: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Zs2)),append(A,Ys,Qs))),lexord(A,R))
     => ( ( Xs != Ys )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
         => ( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(A),nat,size_size(list(A)),Qs) )
           => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R)) ) ) ) ) ).

% lexord_sufE
tff(fact_8118_lexord__lex,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Y)),lex(A,R))
    <=> ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Y)),lexord(A,R))
        & ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_8119_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R))
     => ~ ! [X2: A,Y4: A] :
            ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4)),R)
           => ! [Us2: list(A),Vs2: list(A)] :
                ( ( Xs = append(A,Us2,cons(A,X2,Vs2)) )
               => ( Ys != append(A,Us2,cons(A,Y4,Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_8120_listrel1I,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),R)
     => ( ( Xs = append(A,Us,cons(A,Xb,Vs)) )
       => ( ( Ys = append(A,Us,cons(A,Y,Vs)) )
         => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)) ) ) ) ).

% listrel1I
tff(fact_8121_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),transitive_rtrancl(A,R))
     => ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R)))
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),cons(A,Xb,Xs)),cons(A,Y,Ys))),transitive_rtrancl(list(A),listrel1(A,R))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_8122_lexord__append__left__rightI,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),U: list(A),Xb: list(A),Y: list(A)] :
      ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R)
     => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,U,cons(A,A2,Xb))),append(A,U,cons(A,B2,Y)))),lexord(A,R)) ) ).

% lexord_append_left_rightI
tff(fact_8123_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs2: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs2))),lexord(A,R))
    <=> ( ? [X: A] :
            ( aa(set(A),$o,member(A,X),aa(list(A),set(A),set2(A),Xs))
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),R) )
        | aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs2)),lexord(A,R)) ) ) ).

% lexord_same_pref_iff
tff(fact_8124_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W: list(A),R: set(product_prod(A,A)),V: list(A),Z: list(A)] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W)),lexord(A,R))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W)),aa(list(A),nat,size_size(list(A)),U))
       => aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,U,V)),append(A,W,Z))),lexord(A,R)) ) ) ).

% lexord_sufI
tff(fact_8125_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,cons(A,Xb,nil(A)))),append(A,Ys,cons(A,Y,nil(A))))),listrel1(A,R))
    <=> ( ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R))
          & ( Xb = Y ) )
        | ( ( Xs = Ys )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y)),R) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_8126_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R))
    <=> ? [Y3: A,N4: nat] :
          ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N4)),Y3)),R)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
          & ( Ys = list_update(A,Xs,N4,Y3) ) ) ) ).

% listrel1_iff_update
tff(fact_8127_List_Olexordp__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( lexordp(A,R,Xs,Ys)
    <=> aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R)))) ) ).

% List.lexordp_def
tff(fact_8128_listrel1p__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( listrel1p(A,R,Xs,Ys)
    <=> aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R)))) ) ).

% listrel1p_def
tff(fact_8129_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: set(C),F2: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A3,F2,G) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_bbd(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),A3),F2),G)) ).

% image2_def
tff(fact_8130_and__not__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: option(num)] :
      ( ( bit_and_not_num(Xb,Xaa) = Y )
     => ( ( ( Xb = one2 )
         => ( ( Xaa = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( Xb = one2 )
           => ( ? [N: num] : Xaa = aa(num,num,bit0,N)
             => ( Y != aa(num,option(num),some(num),one2) ) ) )
         => ( ( ( Xb = one2 )
             => ( ? [N: num] : Xaa = aa(num,num,bit1,N)
               => ( Y != none(num) ) ) )
           => ( ! [M: num] :
                  ( ( Xb = aa(num,num,bit0,M) )
                 => ( ( Xaa = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M)) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa(num,num,bit0,M) )
                   => ! [N: num] :
                        ( ( Xaa = aa(num,num,bit0,N) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M,N)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa(num,num,bit0,M) )
                     => ! [N: num] :
                          ( ( Xaa = aa(num,num,bit1,N) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M,N)) ) ) )
                 => ( ! [M: num] :
                        ( ( Xb = aa(num,num,bit1,M) )
                       => ( ( Xaa = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa(num,num,bit1,M) )
                         => ! [N: num] :
                              ( ( Xaa = aa(num,num,bit0,N) )
                             => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_ayh(num,option(num)),bit_and_not_num(M,N)) ) ) )
                     => ~ ! [M: num] :
                            ( ( Xb = aa(num,num,bit1,M) )
                           => ! [N: num] :
                                ( ( Xaa = aa(num,num,bit1,N) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M,N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_8131_map__option__eq__Some,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),Xo: option(B),Y: A] :
      ( ( aa(option(B),option(A),map_option(B,A,F2),Xo) = aa(A,option(A),some(A),Y) )
    <=> ? [Z3: B] :
          ( ( Xo = aa(B,option(B),some(B),Z3) )
          & ( aa(B,A,F2,Z3) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_8132_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xb: option(B)] :
      ( ( none(A) = aa(option(B),option(A),map_option(B,A,F2),Xb) )
    <=> ( Xb = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_8133_map__option__is__None,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Opt: option(B)] :
      ( ( aa(option(B),option(A),map_option(B,A,F2),Opt) = none(A) )
    <=> ( Opt = none(B) ) ) ).

% map_option_is_None
tff(fact_8134_option_Omap__disc__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A2: option(B)] :
      ( ( aa(option(B),option(A),map_option(B,A,F2),A2) = none(A) )
    <=> ( A2 = none(B) ) ) ).

% option.map_disc_iff
tff(fact_8135_case__map__option,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: A,H: fun(B,A),F2: fun(C,B),Xb: option(C)] : case_option(A,B,G,H,aa(option(C),option(B),map_option(C,B,F2),Xb)) = case_option(A,C,G,aa(fun(C,B),fun(C,A),comp(B,A,C,H),F2),Xb) ).

% case_map_option
tff(fact_8136_option_Osimps_I9_J,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),X23: B] : aa(option(B),option(A),map_option(B,A,F2),aa(B,option(B),some(B),X23)) = aa(A,option(A),some(A),aa(B,A,F2,X23)) ).

% option.simps(9)
tff(fact_8137_map__option__cong,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),Y: option(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( Xb = Y )
     => ( ! [A4: A] :
            ( ( Y = aa(A,option(A),some(A),A4) )
           => ( aa(A,B,F2,A4) = aa(A,B,G,A4) ) )
       => ( aa(option(A),option(B),map_option(A,B,F2),Xb) = aa(option(A),option(B),map_option(A,B,G),Y) ) ) ) ).

% map_option_cong
tff(fact_8138_option_Omap__ident,axiom,
    ! [A: $tType,Ta: option(A)] : aa(option(A),option(A),map_option(A,A,aTP_Lamp_ml(A,A)),Ta) = Ta ).

% option.map_ident
tff(fact_8139_option_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => inj_on(option(A),option(B),map_option(A,B,F2),top_top(set(option(A)))) ) ).

% option.inj_map
tff(fact_8140_option_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,nat),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),comp(option(B),nat,option(A),size_option(B,F2)),map_option(A,B,G)) = size_option(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F2),G)) ).

% option.size_gen_o_map
tff(fact_8141_map__option_Ocomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),G: fun(A,C)] : aa(fun(option(A),option(C)),fun(option(A),option(B)),comp(option(C),option(B),option(A),map_option(C,B,F2)),map_option(A,C,G)) = map_option(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,F2),G)) ).

% map_option.comp
tff(fact_8142_option_Omap__comp,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(B,A),F2: fun(C,B),V: option(C)] : aa(option(B),option(A),map_option(B,A,G),aa(option(C),option(B),map_option(C,B,F2),V)) = aa(option(C),option(A),map_option(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,G),F2)),V) ).

% option.map_comp
tff(fact_8143_map__option_Ocompositionality,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),Option: option(C)] : aa(option(B),option(A),map_option(B,A,F2),aa(option(C),option(B),map_option(C,B,G),Option)) = aa(option(C),option(A),map_option(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F2),G)),Option) ).

% map_option.compositionality
tff(fact_8144_and__not__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa(num,num,bit0,Mb),aa(num,num,bit0,Nb)) = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(5)
tff(fact_8145_and__not__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa(num,num,bit0,Mb),aa(num,num,bit1,Nb)) = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(6)
tff(fact_8146_and__not__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa(num,num,bit1,Mb),aa(num,num,bit1,Nb)) = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(9)
tff(fact_8147_option_Osimps_I8_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(option(B),option(A),map_option(B,A,F2),none(B)) = none(A) ).

% option.simps(8)
tff(fact_8148_option_Omap__sel,axiom,
    ! [B: $tType,A: $tType,A2: option(A),F2: fun(A,B)] :
      ( ( A2 != none(A) )
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),map_option(A,B,F2),A2)) = aa(A,B,F2,aa(option(A),A,the2(A),A2)) ) ) ).

% option.map_sel
tff(fact_8149_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F2: fun(B,A),Xb: B,C2: C,G: fun(B,C),A3: set(B)] :
      ( ( B2 = aa(B,A,F2,Xb) )
     => ( ( C2 = aa(B,C,G,Xb) )
       => ( aa(set(B),$o,member(B,Xb),A3)
         => aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C2)),bNF_Greatest_image2(B,A,C,A3,F2,G)) ) ) ) ).

% image2_eqI
tff(fact_8150_map__option__case,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Y: option(B)] : aa(option(B),option(A),map_option(B,A,F2),Y) = case_option(option(A),B,none(A),aTP_Lamp_bbe(fun(B,A),fun(B,option(A)),F2),Y) ).

% map_option_case
tff(fact_8151_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,B),X4: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F2)),aTP_Lamp_bbf(A,option(C))),X4) = none(B) ).

% map_option_o_empty
tff(fact_8152_and__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Xb),Xaa) = Y )
     => ( ( ( Xb = one2 )
         => ( ( Xaa = one2 )
           => ( Y != aa(num,option(num),some(num),one2) ) ) )
       => ( ( ( Xb = one2 )
           => ( ? [N: num] : Xaa = aa(num,num,bit0,N)
             => ( Y != none(num) ) ) )
         => ( ( ( Xb = one2 )
             => ( ? [N: num] : Xaa = aa(num,num,bit1,N)
               => ( Y != aa(num,option(num),some(num),one2) ) ) )
           => ( ( ? [M: num] : Xb = aa(num,num,bit0,M)
               => ( ( Xaa = one2 )
                 => ( Y != none(num) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa(num,num,bit0,M) )
                   => ! [N: num] :
                        ( ( Xaa = aa(num,num,bit0,N) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa(num,num,bit0,M) )
                     => ! [N: num] :
                          ( ( Xaa = aa(num,num,bit1,N) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ) ) )
                 => ( ( ? [M: num] : Xb = aa(num,num,bit1,M)
                     => ( ( Xaa = one2 )
                       => ( Y != aa(num,option(num),some(num),one2) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa(num,num,bit1,M) )
                         => ! [N: num] :
                              ( ( Xaa = aa(num,num,bit0,N) )
                             => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ) ) )
                     => ~ ! [M: num] :
                            ( ( Xb = aa(num,num,bit1,M) )
                           => ! [N: num] :
                                ( ( Xaa = aa(num,num,bit1,N) )
                               => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_ayh(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_8153_and__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit0,Mb)),aa(num,num,bit0,Nb)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(5)
tff(fact_8154_and__num_Osimps_I1_J,axiom,
    aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,one2),one2) = aa(num,option(num),some(num),one2) ).

% and_num.simps(1)
tff(fact_8155_and__num_Osimps_I3_J,axiom,
    ! [Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,one2),aa(num,num,bit1,Nb)) = aa(num,option(num),some(num),one2) ).

% and_num.simps(3)
tff(fact_8156_and__num_Osimps_I7_J,axiom,
    ! [Mb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit1,Mb)),one2) = aa(num,option(num),some(num),one2) ).

% and_num.simps(7)
tff(fact_8157_and__num_Osimps_I2_J,axiom,
    ! [Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,one2),aa(num,num,bit0,Nb)) = none(num) ).

% and_num.simps(2)
tff(fact_8158_and__num_Osimps_I4_J,axiom,
    ! [Mb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit0,Mb)),one2) = none(num) ).

% and_num.simps(4)
tff(fact_8159_and__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num,Q2: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb) = aa(num,option(num),some(num),Q2) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% and_num_eq_Some_iff
tff(fact_8160_and__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit0,Mb)),aa(num,num,bit1,Nb)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(6)
tff(fact_8161_and__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit1,Mb)),aa(num,num,bit0,Nb)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(8)
tff(fact_8162_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_8163_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb)) ) ).

% numeral_and_num
tff(fact_8164_and__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit1,Mb)),aa(num,num,bit1,Nb)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_ayh(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(9)
tff(fact_8165_xor__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Xb),Xaa) = Y )
     => ( ( ( Xb = one2 )
         => ( ( Xaa = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( Xb = one2 )
           => ! [N: num] :
                ( ( Xaa = aa(num,num,bit0,N) )
               => ( Y != aa(num,option(num),some(num),aa(num,num,bit1,N)) ) ) )
         => ( ( ( Xb = one2 )
             => ! [N: num] :
                  ( ( Xaa = aa(num,num,bit1,N) )
                 => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,N)) ) ) )
           => ( ! [M: num] :
                  ( ( Xb = aa(num,num,bit0,M) )
                 => ( ( Xaa = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit1,M)) ) ) )
             => ( ! [M: num] :
                    ( ( Xb = aa(num,num,bit0,M) )
                   => ! [N: num] :
                        ( ( Xaa = aa(num,num,bit0,N) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N)) ) ) )
               => ( ! [M: num] :
                      ( ( Xb = aa(num,num,bit0,M) )
                     => ! [N: num] :
                          ( ( Xaa = aa(num,num,bit1,N) )
                         => ( Y != aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N))) ) ) )
                 => ( ! [M: num] :
                        ( ( Xb = aa(num,num,bit1,M) )
                       => ( ( Xaa = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M)) ) ) )
                   => ( ! [M: num] :
                          ( ( Xb = aa(num,num,bit1,M) )
                         => ! [N: num] :
                              ( ( Xaa = aa(num,num,bit0,N) )
                             => ( Y != aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N))) ) ) )
                     => ~ ! [M: num] :
                            ( ( Xb = aa(num,num,bit1,M) )
                           => ! [N: num] :
                                ( ( Xaa = aa(num,num,bit1,N) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_8166_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_8167_xor__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit0,Mb)),aa(num,num,bit0,Nb)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Mb),Nb)) ).

% xor_num.simps(5)
tff(fact_8168_xor__num_Osimps_I1_J,axiom,
    aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,one2),one2) = none(num) ).

% xor_num.simps(1)
tff(fact_8169_xor__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num,Q2: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Mb),Nb) = aa(num,option(num),some(num),Q2) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% xor_num_eq_Some_iff
tff(fact_8170_xor__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit1,Mb)),aa(num,num,bit1,Nb)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Mb),Nb)) ).

% xor_num.simps(9)
tff(fact_8171_xor__num_Osimps_I2_J,axiom,
    ! [Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,one2),aa(num,num,bit0,Nb)) = aa(num,option(num),some(num),aa(num,num,bit1,Nb)) ).

% xor_num.simps(2)
tff(fact_8172_xor__num_Osimps_I3_J,axiom,
    ! [Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,one2),aa(num,num,bit1,Nb)) = aa(num,option(num),some(num),aa(num,num,bit0,Nb)) ).

% xor_num.simps(3)
tff(fact_8173_xor__num_Osimps_I4_J,axiom,
    ! [Mb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit0,Mb)),one2) = aa(num,option(num),some(num),aa(num,num,bit1,Mb)) ).

% xor_num.simps(4)
tff(fact_8174_xor__num_Osimps_I7_J,axiom,
    ! [Mb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit1,Mb)),one2) = aa(num,option(num),some(num),aa(num,num,bit0,Mb)) ).

% xor_num.simps(7)
tff(fact_8175_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Mb),Nb) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_8176_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Mb)),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Mb),Nb)) ) ).

% numeral_xor_num
tff(fact_8177_xor__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit1,Mb)),aa(num,num,bit0,Nb)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Mb),Nb))) ).

% xor_num.simps(8)
tff(fact_8178_xor__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit0,Mb)),aa(num,num,bit1,Nb)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Mb),Nb))) ).

% xor_num.simps(6)
tff(fact_8179_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_8180_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I2: B,B4: set(A),J4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),fun_upd(B,set(A),A3,I2,B4)),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J4),aa(set(B),set(B),insert(B,I2),bot_bot(set(B))))))),
        $ite(aa(set(B),$o,member(B,I2),J4),B4,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_8181_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_ei(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),aa(num,num,bit0,one2)))),one_one(nat)))) ).

% ATP.lambda_1
tff(fact_8182_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_afg(A,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,exp(A),Uu)),one_one(A))),Uu) ) ).

% ATP.lambda_2
tff(fact_8183_ATP_Olambda__3,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_rt(set(set(A)),int),Uu) = 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),plus_plus(nat),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).

% ATP.lambda_3
tff(fact_8184_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_avy(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_4
tff(fact_8185_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_gj(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),aa(num,num,bit0,one2)))),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_5
tff(fact_8186_ATP_Olambda__6,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_ma(real,$o),Uu)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_6
tff(fact_8187_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ago(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_7
tff(fact_8188_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_eh(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_8
tff(fact_8189_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_agp(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_9
tff(fact_8190_ATP_Olambda__10,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_afk(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,Uu)),sin(real,Uu)) ).

% ATP.lambda_10
tff(fact_8191_ATP_Olambda__11,axiom,
    ! [A: $tType] :
      ( topolo4211221413907600880p_mult(A)
     => ! [Uu: product_prod(A,A)] : aa(product_prod(A,A),A,aTP_Lamp_art(product_prod(A,A),A),Uu) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(product_prod(A,A),A,product_fst(A,A),Uu)),aa(product_prod(A,A),A,product_snd(A,A),Uu)) ) ).

% ATP.lambda_11
tff(fact_8192_ATP_Olambda__12,axiom,
    ! [A: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [Uu: product_prod(A,A)] : aa(product_prod(A,A),A,aTP_Lamp_aru(product_prod(A,A),A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(product_prod(A,A),A,product_fst(A,A),Uu)),aa(product_prod(A,A),A,product_snd(A,A),Uu)) ) ).

% ATP.lambda_12
tff(fact_8193_ATP_Olambda__13,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_ajv(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Uu)),Uu) ).

% ATP.lambda_13
tff(fact_8194_ATP_Olambda__14,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( aa(nat,$o,aTP_Lamp_qz(nat,$o),Uu)
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_14
tff(fact_8195_ATP_Olambda__15,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_afx(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_15
tff(fact_8196_ATP_Olambda__16,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_vf(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_16
tff(fact_8197_ATP_Olambda__17,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_va(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).

% ATP.lambda_17
tff(fact_8198_ATP_Olambda__18,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_uz(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).

% ATP.lambda_18
tff(fact_8199_ATP_Olambda__19,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_il(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_19
tff(fact_8200_ATP_Olambda__20,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_abq(A,list(A)),Uu) = cons(A,Uu,nil(A)) ).

% ATP.lambda_20
tff(fact_8201_ATP_Olambda__21,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_sb(A,set(A)),Uu) = aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))) ).

% ATP.lambda_21
tff(fact_8202_ATP_Olambda__22,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_agh(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_22
tff(fact_8203_ATP_Olambda__23,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_agn(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_23
tff(fact_8204_ATP_Olambda__24,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_axl(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).

% ATP.lambda_24
tff(fact_8205_ATP_Olambda__25,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A] : aa(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_atg(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),Uu) = aa(fun(B,fun(C,product_prod(product_prod(A,B),C))),fun(product_prod(B,C),product_prod(product_prod(A,B),C)),product_case_prod(B,C,product_prod(product_prod(A,B),C)),aTP_Lamp_atf(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).

% ATP.lambda_25
tff(fact_8206_ATP_Olambda__26,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_xn(real,real),Uu) = suminf(real,aTP_Lamp_ee(real,fun(nat,real),Uu)) ).

% ATP.lambda_26
tff(fact_8207_ATP_Olambda__27,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_ra(nat,set(nat)),Uu) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bh(nat,fun(nat,$o),Uu)) ).

% ATP.lambda_27
tff(fact_8208_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real] : aa(real,filter(A),aTP_Lamp_ajl(real,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ajk(real,fun(A,$o),Uu))) ) ).

% ATP.lambda_28
tff(fact_8209_ATP_Olambda__29,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_atl(fun(A,B),set(product_prod(A,B))),Uu) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_atk(fun(A,B),fun(A,fun(B,$o)),Uu))) ).

% ATP.lambda_29
tff(fact_8210_ATP_Olambda__30,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(complex,complex)),aTP_Lamp_arl(real,filter(product_prod(complex,complex))),Uu) = aa(set(product_prod(complex,complex)),filter(product_prod(complex,complex)),principal(product_prod(complex,complex)),aa(fun(product_prod(complex,complex),$o),set(product_prod(complex,complex)),collect(product_prod(complex,complex)),aa(fun(complex,fun(complex,$o)),fun(product_prod(complex,complex),$o),product_case_prod(complex,complex,$o),aTP_Lamp_ark(real,fun(complex,fun(complex,$o)),Uu)))) ).

% ATP.lambda_30
tff(fact_8211_ATP_Olambda__31,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(real,real)),aTP_Lamp_arj(real,filter(product_prod(real,real))),Uu) = aa(set(product_prod(real,real)),filter(product_prod(real,real)),principal(product_prod(real,real)),aa(fun(product_prod(real,real),$o),set(product_prod(real,real)),collect(product_prod(real,real)),aa(fun(real,fun(real,$o)),fun(product_prod(real,real),$o),product_case_prod(real,real,$o),aTP_Lamp_ari(real,fun(real,fun(real,$o)),Uu)))) ).

% ATP.lambda_31
tff(fact_8212_ATP_Olambda__32,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real] : aa(real,filter(product_prod(A,A)),aTP_Lamp_ard(real,filter(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),filter(product_prod(A,A)),principal(product_prod(A,A)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_arc(real,fun(A,fun(A,$o)),Uu)))) ) ).

% ATP.lambda_32
tff(fact_8213_ATP_Olambda__33,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_agj(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_33
tff(fact_8214_ATP_Olambda__34,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_lm(nat,A),Uu) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu)) ) ).

% ATP.lambda_34
tff(fact_8215_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_agb(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_35
tff(fact_8216_ATP_Olambda__36,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_aqp(A,B),Uu) = ring_1_of_int(B,archim6421214686448440834_floor(A,Uu)) ) ).

% ATP.lambda_36
tff(fact_8217_ATP_Olambda__37,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_aqq(A,B),Uu) = ring_1_of_int(B,archimedean_ceiling(A,Uu)) ) ).

% ATP.lambda_37
tff(fact_8218_ATP_Olambda__38,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_ayh(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_38
tff(fact_8219_ATP_Olambda__39,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_ayd(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).

% ATP.lambda_39
tff(fact_8220_ATP_Olambda__40,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_nf(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_40
tff(fact_8221_ATP_Olambda__41,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ng(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_41
tff(fact_8222_ATP_Olambda__42,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bl(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_42
tff(fact_8223_ATP_Olambda__43,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_ajd(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_greaterThan(A),Uu)) ) ).

% ATP.lambda_43
tff(fact_8224_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_ajc(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_lessThan(A),Uu)) ) ).

% ATP.lambda_44
tff(fact_8225_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_aof(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_45
tff(fact_8226_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_aog(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_46
tff(fact_8227_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_ajh(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_47
tff(fact_8228_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_aji(A,filter(A)),Uu) = aa(set(A),filter(A),principal(A),aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_48
tff(fact_8229_ATP_Olambda__49,axiom,
    ! [Uu: int] : aa(int,nat,aTP_Lamp_rb(int,nat),Uu) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_49
tff(fact_8230_ATP_Olambda__50,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_azg(product_prod(A,A),$o),Uu)
        <=> ? [X: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X) ) ) ).

% ATP.lambda_50
tff(fact_8231_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_azp(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y3: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) ) ) ) ).

% ATP.lambda_51
tff(fact_8232_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_azq(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y3: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) ) ) ) ).

% ATP.lambda_52
tff(fact_8233_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_azo(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y3: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) ) ) ) ).

% ATP.lambda_53
tff(fact_8234_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_azn(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y3: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X) ) ) ) ).

% ATP.lambda_54
tff(fact_8235_ATP_Olambda__55,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_ayu(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y3: A] :
              ( ( Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3) )
              & ( X != Y3 ) ) ) ) ).

% ATP.lambda_55
tff(fact_8236_ATP_Olambda__56,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_ayg(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_56
tff(fact_8237_ATP_Olambda__57,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_bai(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_bag(nat,fun(num,option(num)),Uua),aTP_Lamp_bah(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_57
tff(fact_8238_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_lu(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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_58
tff(fact_8239_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_ip(nat,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_59
tff(fact_8240_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_lt(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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_60
tff(fact_8241_ATP_Olambda__61,axiom,
    ! [Uu: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_gk(fun(nat,real),fun(nat,real),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),zero_zero(real),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% ATP.lambda_61
tff(fact_8242_ATP_Olambda__62,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_mi(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(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),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_62
tff(fact_8243_ATP_Olambda__63,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_io(nat,fun(nat,A),Uu),Uua) = $ite(~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_63
tff(fact_8244_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(nat,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahz(fun(nat,A),fun(A,$o),Uu),Uua)
        <=> filterlim(nat,A,Uu,topolo7230453075368039082e_nhds(A,Uua),at_top(nat)) ) ) ).

% ATP.lambda_64
tff(fact_8245_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_asj(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),top_top(A)) ) ).

% ATP.lambda_65
tff(fact_8246_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_uv(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ).

% ATP.lambda_66
tff(fact_8247_ATP_Olambda__67,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_ase(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ).

% ATP.lambda_67
tff(fact_8248_ATP_Olambda__68,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_baj(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_bai(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_68
tff(fact_8249_ATP_Olambda__69,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_bag(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_ayd(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_69
tff(fact_8250_ATP_Olambda__70,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_aye(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_ayd(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_70
tff(fact_8251_ATP_Olambda__71,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_qc(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu),Uua)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_71
tff(fact_8252_ATP_Olambda__72,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_jg(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_jf(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_72
tff(fact_8253_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_je(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_jd(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_73
tff(fact_8254_ATP_Olambda__74,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_dn(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,aa(real,fun(real,real),minus_minus(real),Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_74
tff(fact_8255_ATP_Olambda__75,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ga(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),aa(num,num,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),aa(num,num,bit0,one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_75
tff(fact_8256_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_gm(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),aa(num,num,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),aa(num,num,bit0,one2))),Uua))) ).

% ATP.lambda_76
tff(fact_8257_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_kd(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_77
tff(fact_8258_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_lh(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),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_78
tff(fact_8259_ATP_Olambda__79,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_lw(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_79
tff(fact_8260_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_lv(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),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),aa(num,num,bit0,one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_80
tff(fact_8261_ATP_Olambda__81,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_pw(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
        l2: int,
        l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),code_int_of_integer(Uu)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_81
tff(fact_8262_ATP_Olambda__82,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_dl(nat,fun(nat,a)),Uu),Uua) = $let(
        m3: a,
        m3:= aa(a,a,aa(a,fun(a,a),times_times(a),aa(num,a,numeral_numeral(a),aa(num,num,bit0,one2))),aa(nat,a,semiring_1_of_nat(a),Uu)),
        $ite(Uua = zero_zero(nat),m3,aa(a,a,aa(a,fun(a,a),plus_plus(a),m3),one_one(a))) ) ).

% ATP.lambda_82
tff(fact_8263_ATP_Olambda__83,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_mf(complex,fun(real,$o),Uu),Uua)
    <=> ( ( aa(complex,complex,sgn_sgn(complex),Uu) = cis(Uua) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi) ) ) ).

% ATP.lambda_83
tff(fact_8264_ATP_Olambda__84,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_md(real,fun(int,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),ring_1_of_int(real,Uua)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu),ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_84
tff(fact_8265_ATP_Olambda__85,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_me(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),ring_1_of_int(rat,Uua)),Uu)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu),ring_1_of_int(rat,aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_85
tff(fact_8266_ATP_Olambda__86,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ee(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),aa(num,num,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),aa(num,num,bit0,one2)))),one_one(nat))))) ).

% ATP.lambda_86
tff(fact_8267_ATP_Olambda__87,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_xo(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),aa(num,num,bit0,one2))))) ).

% ATP.lambda_87
tff(fact_8268_ATP_Olambda__88,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_afl(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_88
tff(fact_8269_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_kh(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_89
tff(fact_8270_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kz(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_90
tff(fact_8271_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_kq(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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_91
tff(fact_8272_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lc(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_92
tff(fact_8273_ATP_Olambda__93,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ql(nat,fun(nat,$o)),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_93
tff(fact_8274_ATP_Olambda__94,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_ru(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_94
tff(fact_8275_ATP_Olambda__95,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_vi(set(option(A)),fun(option(A),$o),Uu),Uua)
    <=> ( aa(set(option(A)),$o,member(option(A),Uua),Uu)
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_95
tff(fact_8276_ATP_Olambda__96,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jx(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),aa(num,num,bit0,one2))) ).

% ATP.lambda_96
tff(fact_8277_ATP_Olambda__97,axiom,
    ! [Uu: set(int),Uua: int] :
      ( aa(int,$o,aTP_Lamp_bab(set(int),fun(int,$o),Uu),Uua)
    <=> ( aa(set(int),$o,member(int,Uua),Uu)
        & ! [X: int] :
            ( aa(set(int),$o,member(int,X),Uu)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Uua) ) ) ) ).

% ATP.lambda_97
tff(fact_8278_ATP_Olambda__98,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_bae(set(set(A)),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,member(set(A),Uua),Uu)
        & ! [X: set(A)] :
            ( aa(set(set(A)),$o,member(set(A),X),Uu)
           => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),X) ) ) ) ).

% ATP.lambda_98
tff(fact_8279_ATP_Olambda__99,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ej(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_99
tff(fact_8280_ATP_Olambda__100,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_ajz(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_100
tff(fact_8281_ATP_Olambda__101,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_axe(nat,fun(nat,set(nat)),Uu),Uua) = set_or3652927894154168847AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua),Uu) ).

% ATP.lambda_101
tff(fact_8282_ATP_Olambda__102,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_asn(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uua)),Uua) ).

% ATP.lambda_102
tff(fact_8283_ATP_Olambda__103,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ja(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_103
tff(fact_8284_ATP_Olambda__104,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_iz(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_104
tff(fact_8285_ATP_Olambda__105,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_axg(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_105
tff(fact_8286_ATP_Olambda__106,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_avo(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_106
tff(fact_8287_ATP_Olambda__107,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_da(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_107
tff(fact_8288_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ao(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_108
tff(fact_8289_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_mh(A,fun(A,$o),Uu),Uua)
        <=> ( aa(set(A),$o,member(A,Uua),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu) ) ) ) ).

% ATP.lambda_109
tff(fact_8290_ATP_Olambda__110,axiom,
    ! [A: $tType,Uu: fun(set(A),$o),Uua: set(A)] :
      ( aa(set(A),$o,aa(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_azk(fun(set(A),$o),fun(set(A),$o)),Uu),Uua)
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A14: set(A),A5: A] :
            ( ( Uua = aa(set(A),set(A),insert(A,A5),A14) )
            & aa(set(A),$o,Uu,A14) ) ) ) ).

% ATP.lambda_110
tff(fact_8291_ATP_Olambda__111,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agx(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_111
tff(fact_8292_ATP_Olambda__112,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_aid(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_112
tff(fact_8293_ATP_Olambda__113,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_akb(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_113
tff(fact_8294_ATP_Olambda__114,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_dm(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),aa(num,num,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),aa(num,num,bit0,one2)))),one_one(nat)))) ).

% ATP.lambda_114
tff(fact_8295_ATP_Olambda__115,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_lz(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi)
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_115
tff(fact_8296_ATP_Olambda__116,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_avx(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ) ).

% ATP.lambda_116
tff(fact_8297_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_awc(nat,fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu)) ) ) ).

% ATP.lambda_117
tff(fact_8298_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ih(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),aa(num,num,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),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_118
tff(fact_8299_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_de(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),aa(num,num,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),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_119
tff(fact_8300_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_age(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_120
tff(fact_8301_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_db(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_121
tff(fact_8302_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_og(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_122
tff(fact_8303_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_oe(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_123
tff(fact_8304_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fl(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_124
tff(fact_8305_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dq(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_125
tff(fact_8306_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cs(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_126
tff(fact_8307_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fk(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_127
tff(fact_8308_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( ab_group_add(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),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_128
tff(fact_8309_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dc(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(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_129
tff(fact_8310_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_130
tff(fact_8311_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_du(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_131
tff(fact_8312_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aoc(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y3: A] :
                ( aa(A,$o,Uu,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Uua) ) ) ) ) ).

% ATP.lambda_132
tff(fact_8313_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_iw(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_133
tff(fact_8314_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_it(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)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_134
tff(fact_8315_ATP_Olambda__135,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_amr(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_135
tff(fact_8316_ATP_Olambda__136,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_zt(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(B,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_136
tff(fact_8317_ATP_Olambda__137,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_gi(fun(A,real),fun(A,complex),Uu),Uua) = complex2(aa(A,real,Uu,Uua),zero_zero(real)) ).

% ATP.lambda_137
tff(fact_8318_ATP_Olambda__138,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_ss(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),insert(A,aa(B,A,Uu,Uua)),bot_bot(set(A))) ).

% ATP.lambda_138
tff(fact_8319_ATP_Olambda__139,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_cq(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_139
tff(fact_8320_ATP_Olambda__140,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_hz(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_140
tff(fact_8321_ATP_Olambda__141,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ahh(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_141
tff(fact_8322_ATP_Olambda__142,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ahg(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_afl(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))) ).

% ATP.lambda_142
tff(fact_8323_ATP_Olambda__143,axiom,
    ! [Uu: code_integer,Uua: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_vq(code_integer,fun($o,char)),Uu),(Uua)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aTP_Lamp_vp($o,fun(code_integer,fun($o,char)),(Uua))),code_bit_cut_integer(Uu)) ).

% ATP.lambda_143
tff(fact_8324_ATP_Olambda__144,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_atw(nat,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uu) ).

% ATP.lambda_144
tff(fact_8325_ATP_Olambda__145,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_aht(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_145
tff(fact_8326_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ll(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),aa(num,num,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),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_146
tff(fact_8327_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lr(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_147
tff(fact_8328_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lo(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_148
tff(fact_8329_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ln(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_149
tff(fact_8330_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(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,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_150
tff(fact_8331_ATP_Olambda__151,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_ayc(num,fun(num,int),Uu),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_151
tff(fact_8332_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lb(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_152
tff(fact_8333_ATP_Olambda__153,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_uw(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,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_153
tff(fact_8334_ATP_Olambda__154,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,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_154
tff(fact_8335_ATP_Olambda__155,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) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_155
tff(fact_8336_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ahc(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_156
tff(fact_8337_ATP_Olambda__157,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ahb(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_157
tff(fact_8338_ATP_Olambda__158,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),sin_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_158
tff(fact_8339_ATP_Olambda__159,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fw(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_159
tff(fact_8340_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_ait(A,fun(set(A),$o),Uu),Uua)
        <=> ( aa(set(A),$o,topolo1002775350975398744n_open(A),Uua)
            & aa(set(A),$o,member(A,Uu),Uua) ) ) ) ).

% ATP.lambda_160
tff(fact_8341_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qq(nat,fun(nat,A)),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_161
tff(fact_8342_ATP_Olambda__162,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_px(code_integer,fun(code_integer,num)),Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ).

% ATP.lambda_162
tff(fact_8343_ATP_Olambda__163,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_py(code_integer,fun(code_integer,nat)),Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ).

% ATP.lambda_163
tff(fact_8344_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ld(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_164
tff(fact_8345_ATP_Olambda__165,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_lx(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,real,exp(real),Uua) = Uu ) ) ).

% ATP.lambda_165
tff(fact_8346_ATP_Olambda__166,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_bb(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),aa(num,num,bit0,one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_166
tff(fact_8347_ATP_Olambda__167,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_bc(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),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_167
tff(fact_8348_ATP_Olambda__168,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ha(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),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_168
tff(fact_8349_ATP_Olambda__169,axiom,
    ! [A: $tType,Uu: A,Uua: set(set(A))] : aa(set(set(A)),set(set(A)),aa(A,fun(set(set(A)),set(set(A))),aTP_Lamp_axf(A,fun(set(set(A)),set(set(A)))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),Uua),aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,Uu)),Uua)) ).

% ATP.lambda_169
tff(fact_8350_ATP_Olambda__170,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kg(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_170
tff(fact_8351_ATP_Olambda__171,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_mk(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),insert(A,Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_171
tff(fact_8352_ATP_Olambda__172,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agz(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_172
tff(fact_8353_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ags(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_173
tff(fact_8354_ATP_Olambda__174,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_ati(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_174
tff(fact_8355_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ok(A,fun(nat,A),Uu),Uua) = bit_se4730199178511100633sh_bit(A,Uua,aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_175
tff(fact_8356_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_agt(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_176
tff(fact_8357_ATP_Olambda__177,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_bo(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_177
tff(fact_8358_ATP_Olambda__178,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agu(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_178
tff(fact_8359_ATP_Olambda__179,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agk(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_179
tff(fact_8360_ATP_Olambda__180,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_tb(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,Uu),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ).

% ATP.lambda_180
tff(fact_8361_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tk(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_181
tff(fact_8362_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agl(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_182
tff(fact_8363_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agv(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_183
tff(fact_8364_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_184
tff(fact_8365_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_agw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_185
tff(fact_8366_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ajk(real,fun(A,$o),Uu),Uua)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu),real_V7770717601297561774m_norm(A,Uua)) ) ) ).

% ATP.lambda_186
tff(fact_8367_ATP_Olambda__187,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_awb(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ).

% ATP.lambda_187
tff(fact_8368_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_afy(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_188
tff(fact_8369_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lg(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_189
tff(fact_8370_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_oo(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_190
tff(fact_8371_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_if(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_191
tff(fact_8372_ATP_Olambda__192,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_alc(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> eventually(A,Uu,topolo7230453075368039082e_nhds(A,Uua)) ) ) ).

% ATP.lambda_192
tff(fact_8373_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat,Uua: A] : aa(A,A,aTP_Lamp_abh(nat,fun(A,A),Uu),Uua) = comm_s3205402744901411588hammer(A,Uua,Uu) ) ).

% ATP.lambda_193
tff(fact_8374_ATP_Olambda__194,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: real] : aa(real,A,aTP_Lamp_aho(A,fun(real,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),Uu) ) ).

% ATP.lambda_194
tff(fact_8375_ATP_Olambda__195,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_an(set(A),fun(set(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ).

% ATP.lambda_195
tff(fact_8376_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_al(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_196
tff(fact_8377_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_awx(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_197
tff(fact_8378_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_pz(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_198
tff(fact_8379_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_iq(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_199
tff(fact_8380_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_us(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_200
tff(fact_8381_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahm(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_201
tff(fact_8382_ATP_Olambda__202,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mp(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_202
tff(fact_8383_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ass(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_203
tff(fact_8384_ATP_Olambda__204,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_am(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_204
tff(fact_8385_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_awy(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_205
tff(fact_8386_ATP_Olambda__206,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_akk(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_206
tff(fact_8387_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_dr(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_207
tff(fact_8388_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_afu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_208
tff(fact_8389_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahp(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_209
tff(fact_8390_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ac(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_210
tff(fact_8391_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nk(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_211
tff(fact_8392_ATP_Olambda__212,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_atx(real,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),minus_minus(real),Uua),Uu) ).

% ATP.lambda_212
tff(fact_8393_ATP_Olambda__213,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ol(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_213
tff(fact_8394_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_att(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_214
tff(fact_8395_ATP_Olambda__215,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mn(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_215
tff(fact_8396_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_nr(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_216
tff(fact_8397_ATP_Olambda__217,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_wt(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_217
tff(fact_8398_ATP_Olambda__218,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_awm(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_218
tff(fact_8399_ATP_Olambda__219,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_ue(set(A),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),Uu) ).

% ATP.lambda_219
tff(fact_8400_ATP_Olambda__220,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_tr(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).

% ATP.lambda_220
tff(fact_8401_ATP_Olambda__221,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_aty(real,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uua),Uu) ).

% ATP.lambda_221
tff(fact_8402_ATP_Olambda__222,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_os(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_222
tff(fact_8403_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_atv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_223
tff(fact_8404_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_asv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_224
tff(fact_8405_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mm(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_225
tff(fact_8406_ATP_Olambda__226,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_ww(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_226
tff(fact_8407_ATP_Olambda__227,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_bh(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uua),Uu) ) ).

% ATP.lambda_227
tff(fact_8408_ATP_Olambda__228,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_bg(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uua),Uu) ) ).

% ATP.lambda_228
tff(fact_8409_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_be(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uua),Uu) ) ) ).

% ATP.lambda_229
tff(fact_8410_ATP_Olambda__230,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_auy(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).

% ATP.lambda_230
tff(fact_8411_ATP_Olambda__231,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_qp(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uu) ).

% ATP.lambda_231
tff(fact_8412_ATP_Olambda__232,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_vd(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),Uu) ).

% ATP.lambda_232
tff(fact_8413_ATP_Olambda__233,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_ve(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_233
tff(fact_8414_ATP_Olambda__234,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ir(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_234
tff(fact_8415_ATP_Olambda__235,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_atm(list(A),fun(A,list(A))),Uu),Uua) = cons(A,Uua,Uu) ).

% ATP.lambda_235
tff(fact_8416_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : aa(A,A,aTP_Lamp_aua(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ).

% ATP.lambda_236
tff(fact_8417_ATP_Olambda__237,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agi(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_237
tff(fact_8418_ATP_Olambda__238,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aoj(set(B),fun(B,$o),Uu),Uua)
    <=> aa(set(B),$o,member(B,Uua),Uu) ) ).

% ATP.lambda_238
tff(fact_8419_ATP_Olambda__239,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_alb(set(A),fun(A,$o),Uu),Uua)
        <=> aa(set(A),$o,member(A,Uua),Uu) ) ) ).

% ATP.lambda_239
tff(fact_8420_ATP_Olambda__240,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ask(set(A),fun(A,$o),Uu),Uua)
        <=> aa(set(A),$o,member(A,Uua),Uu) ) ) ).

% ATP.lambda_240
tff(fact_8421_ATP_Olambda__241,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Uu),Uua)
    <=> aa(set(A),$o,member(A,Uua),Uu) ) ).

% ATP.lambda_241
tff(fact_8422_ATP_Olambda__242,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_avw(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Uua),Uu) ).

% ATP.lambda_242
tff(fact_8423_ATP_Olambda__243,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ek(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_243
tff(fact_8424_ATP_Olambda__244,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ang(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_244
tff(fact_8425_ATP_Olambda__245,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_amw(fun(A,real),fun(A,$o),Uu),Uua)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_245
tff(fact_8426_ATP_Olambda__246,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_anc(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_246
tff(fact_8427_ATP_Olambda__247,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_amq(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_247
tff(fact_8428_ATP_Olambda__248,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_avl(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),Uu) ).

% ATP.lambda_248
tff(fact_8429_ATP_Olambda__249,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ed(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),aa(num,num,bit0,one2))),Uua)),one_one(nat))) ).

% ATP.lambda_249
tff(fact_8430_ATP_Olambda__250,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ec(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),aa(num,num,bit0,one2))),Uua)) ).

% ATP.lambda_250
tff(fact_8431_ATP_Olambda__251,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_akc(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_251
tff(fact_8432_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_afv(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_252
tff(fact_8433_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_afw(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_253
tff(fact_8434_ATP_Olambda__254,axiom,
    ! [Uu: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_anb(fun(real,$o),fun(real,$o),Uu),Uua)
    <=> aa(real,$o,Uu,aa(real,real,inverse_inverse(real),Uua)) ) ).

% ATP.lambda_254
tff(fact_8435_ATP_Olambda__255,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_ajy(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,inverse_inverse(real),Uua)) ).

% ATP.lambda_255
tff(fact_8436_ATP_Olambda__256,axiom,
    ! [Uu: fun(real,real),Uua: nat] : aa(nat,real,aTP_Lamp_asq(fun(real,real),fun(nat,real),Uu),Uua) = aa(real,real,Uu,aa(nat,real,semiring_1_of_nat(real),Uua)) ).

% ATP.lambda_256
tff(fact_8437_ATP_Olambda__257,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(real,A),Uua: nat] : aa(nat,A,aTP_Lamp_aka(fun(real,A),fun(nat,A),Uu),Uua) = aa(real,A,Uu,aa(nat,real,semiring_1_of_nat(real),Uua)) ) ).

% ATP.lambda_257
tff(fact_8438_ATP_Olambda__258,axiom,
    ! [A: $tType,Uu: fun(int,A),Uua: nat] : aa(nat,A,aTP_Lamp_anz(fun(int,A),fun(nat,A),Uu),Uua) = aa(int,A,Uu,aa(nat,int,semiring_1_of_nat(int),Uua)) ).

% ATP.lambda_258
tff(fact_8439_ATP_Olambda__259,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_vw(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_259
tff(fact_8440_ATP_Olambda__260,axiom,
    ! [Uu: fun(real,$o),Uua: real] :
      ( aa(real,$o,aTP_Lamp_amv(fun(real,$o),fun(real,$o),Uu),Uua)
    <=> aa(real,$o,Uu,aa(real,real,uminus_uminus(real),Uua)) ) ).

% ATP.lambda_260
tff(fact_8441_ATP_Olambda__261,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real] : aa(real,A,aTP_Lamp_aig(fun(real,A),fun(real,A),Uu),Uua) = aa(real,A,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_261
tff(fact_8442_ATP_Olambda__262,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_akh(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_262
tff(fact_8443_ATP_Olambda__263,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(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_263
tff(fact_8444_ATP_Olambda__264,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_afq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_264
tff(fact_8445_ATP_Olambda__265,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_265
tff(fact_8446_ATP_Olambda__266,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_266
tff(fact_8447_ATP_Olambda__267,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cv(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_267
tff(fact_8448_ATP_Olambda__268,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_afp(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_268
tff(fact_8449_ATP_Olambda__269,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [Uu: B,Uua: fun(B,A)] : aa(fun(B,A),A,aTP_Lamp_rw(B,fun(fun(B,A),A),Uu),Uua) = aa(B,A,Uua,Uu) ) ).

% ATP.lambda_269
tff(fact_8450_ATP_Olambda__270,axiom,
    ! [A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [Uu: B,Uua: fun(B,A)] : aa(fun(B,A),A,aTP_Lamp_re(B,fun(fun(B,A),A),Uu),Uua) = aa(B,A,Uua,Uu) ) ).

% ATP.lambda_270
tff(fact_8451_ATP_Olambda__271,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_sh(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_271
tff(fact_8452_ATP_Olambda__272,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_rk(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_272
tff(fact_8453_ATP_Olambda__273,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_axx(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_max(A),Uu),Uua)) ) ).

% ATP.lambda_273
tff(fact_8454_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_bas(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_274
tff(fact_8455_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_baw(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_275
tff(fact_8456_ATP_Olambda__276,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_bah(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_276
tff(fact_8457_ATP_Olambda__277,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_ayf(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_277
tff(fact_8458_ATP_Olambda__278,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_bba(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_baz(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_278
tff(fact_8459_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qy(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qx(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_279
tff(fact_8460_ATP_Olambda__280,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qv(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_280
tff(fact_8461_ATP_Olambda__281,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_qu(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_qt(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_281
tff(fact_8462_ATP_Olambda__282,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_qs(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_qr(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_282
tff(fact_8463_ATP_Olambda__283,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qo(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qn(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_283
tff(fact_8464_ATP_Olambda__284,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_xf(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_xe(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_284
tff(fact_8465_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_wu(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_gr(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_285
tff(fact_8466_ATP_Olambda__286,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: B] : aa(B,set(A),aTP_Lamp_anw(fun(A,fun(B,$o)),fun(B,set(A)),Uu),Uua) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_anv(fun(A,fun(B,$o)),fun(B,fun(A,$o)),Uu),Uua)) ).

% ATP.lambda_286
tff(fact_8467_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A] : aa(A,set(A),aTP_Lamp_aio(real,fun(A,set(A)),Uu),Uua) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ain(real,fun(A,fun(A,$o)),Uu),Uua)) ) ).

% ATP.lambda_287
tff(fact_8468_ATP_Olambda__288,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_tx(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_tw(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_288
tff(fact_8469_ATP_Olambda__289,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_ty(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_tu(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_289
tff(fact_8470_ATP_Olambda__290,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_tz(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_tw(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_290
tff(fact_8471_ATP_Olambda__291,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_tv(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_tu(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_291
tff(fact_8472_ATP_Olambda__292,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_ik(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),aa(num,num,bit0,one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua))),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_292
tff(fact_8473_ATP_Olambda__293,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_yt(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_293
tff(fact_8474_ATP_Olambda__294,axiom,
    ! [Uu: fun(real,real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_ake(fun(real,real),fun(real,$o),Uu),Uua)
    <=> ( aa(real,real,Uu,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_294
tff(fact_8475_ATP_Olambda__295,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_avp(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(B),set(set(product_prod(A,B))),image(B,set(product_prod(A,B)),aTP_Lamp_avo(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_295
tff(fact_8476_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_lp(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_296
tff(fact_8477_ATP_Olambda__297,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_bs(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),Uua))) ) ).

% ATP.lambda_297
tff(fact_8478_ATP_Olambda__298,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_la(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_298
tff(fact_8479_ATP_Olambda__299,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_kx(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_299
tff(fact_8480_ATP_Olambda__300,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_ky(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_300
tff(fact_8481_ATP_Olambda__301,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ta(set(set(A)),fun(A,$o),Uu),Uua)
    <=> aa(set($o),$o,complete_Sup_Sup($o),aa(set(set(A)),set($o),image(set(A),$o,member(A,Uua)),Uu)) ) ).

% ATP.lambda_301
tff(fact_8482_ATP_Olambda__302,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ri(set(set(A)),fun(A,$o),Uu),Uua)
    <=> aa(set($o),$o,complete_Inf_Inf($o),aa(set(set(A)),set($o),image(set(A),$o,member(A,Uua)),Uu)) ) ).

% ATP.lambda_302
tff(fact_8483_ATP_Olambda__303,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_um(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_303
tff(fact_8484_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_lj(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_304
tff(fact_8485_ATP_Olambda__305,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_hb(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_305
tff(fact_8486_ATP_Olambda__306,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B)] : aa(set(B),set(A),aTP_Lamp_sn(fun(B,set(A)),fun(set(B),set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),Uu),Uua)) ).

% ATP.lambda_306
tff(fact_8487_ATP_Olambda__307,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_azy(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,Uu),Uua)) ) ).

% ATP.lambda_307
tff(fact_8488_ATP_Olambda__308,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B)] : aa(set(B),set(A),aTP_Lamp_sr(fun(B,set(A)),fun(set(B),set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),Uu),Uua)) ).

% ATP.lambda_308
tff(fact_8489_ATP_Olambda__309,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_azx(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,Uu),Uua)) ) ).

% ATP.lambda_309
tff(fact_8490_ATP_Olambda__310,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(set(A),fun(fun(A,$o),$o),aTP_Lamp_ayl(set(A),fun(fun(A,$o),$o)),Uu),Uua)
    <=> aa(set($o),$o,complete_Inf_Inf($o),aa(set(A),set($o),image(A,$o,Uua),Uu)) ) ).

% ATP.lambda_310
tff(fact_8491_ATP_Olambda__311,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_avv(nat,fun(nat,set(nat)),Uu),Uua) = aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uua)) ).

% ATP.lambda_311
tff(fact_8492_ATP_Olambda__312,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_agr(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_312
tff(fact_8493_ATP_Olambda__313,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wq(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_313
tff(fact_8494_ATP_Olambda__314,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_atf(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = aa(product_prod(A,B),fun(C,product_prod(product_prod(A,B),C)),product_Pair(product_prod(A,B),C),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_314
tff(fact_8495_ATP_Olambda__315,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_ajb(A,fun(A,filter(A)),Uu),Uua) = aa(set(A),filter(A),principal(A),set_or5935395276787703475ssThan(A,Uu,Uua)) ) ).

% ATP.lambda_315
tff(fact_8496_ATP_Olambda__316,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_aja(A,fun(A,filter(A)),Uu),Uua) = aa(set(A),filter(A),principal(A),set_or5935395276787703475ssThan(A,Uua,Uu)) ) ).

% ATP.lambda_316
tff(fact_8497_ATP_Olambda__317,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_axh(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).

% ATP.lambda_317
tff(fact_8498_ATP_Olambda__318,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_axj(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_318
tff(fact_8499_ATP_Olambda__319,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_pp(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu),Uua)) ).

% ATP.lambda_319
tff(fact_8500_ATP_Olambda__320,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_pq(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu)) ).

% ATP.lambda_320
tff(fact_8501_ATP_Olambda__321,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_fx(set(A),fun(A,$o),Uu),Uua)
    <=> ~ aa(set(A),$o,member(A,Uua),Uu) ) ).

% ATP.lambda_321
tff(fact_8502_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_akl(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_322
tff(fact_8503_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ald(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_323
tff(fact_8504_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_akm(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_324
tff(fact_8505_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_akj(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_325
tff(fact_8506_ATP_Olambda__326,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_fq(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_326
tff(fact_8507_ATP_Olambda__327,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_jm(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_327
tff(fact_8508_ATP_Olambda__328,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_es(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_328
tff(fact_8509_ATP_Olambda__329,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_ef(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_329
tff(fact_8510_ATP_Olambda__330,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_hl(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_330
tff(fact_8511_ATP_Olambda__331,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_apg(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_331
tff(fact_8512_ATP_Olambda__332,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_zx(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_332
tff(fact_8513_ATP_Olambda__333,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_hm(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_333
tff(fact_8514_ATP_Olambda__334,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_adb(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_334
tff(fact_8515_ATP_Olambda__335,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: A] : aa(A,B,aTP_Lamp_tj(fun(A,$o),fun(A,B),Uu),Uua) = aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uua)) ) ).

% ATP.lambda_335
tff(fact_8516_ATP_Olambda__336,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,set(B),aTP_Lamp_avm(fun(A,B),fun(A,set(B)),Uu),Uua) = aa(B,set(B),set_ord_atMost(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_336
tff(fact_8517_ATP_Olambda__337,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_agg(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_337
tff(fact_8518_ATP_Olambda__338,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_yu(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_338
tff(fact_8519_ATP_Olambda__339,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_afh(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_339
tff(fact_8520_ATP_Olambda__340,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_apt(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_340
tff(fact_8521_ATP_Olambda__341,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wi(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_341
tff(fact_8522_ATP_Olambda__342,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abf(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_342
tff(fact_8523_ATP_Olambda__343,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_343
tff(fact_8524_ATP_Olambda__344,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ade(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_344
tff(fact_8525_ATP_Olambda__345,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_hd(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_345
tff(fact_8526_ATP_Olambda__346,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_dh(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_346
tff(fact_8527_ATP_Olambda__347,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_ajr(fun(A,nat),fun(A,real),Uu),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_347
tff(fact_8528_ATP_Olambda__348,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_di(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_348
tff(fact_8529_ATP_Olambda__349,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zc(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_349
tff(fact_8530_ATP_Olambda__350,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_apw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_350
tff(fact_8531_ATP_Olambda__351,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aca(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_351
tff(fact_8532_ATP_Olambda__352,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_352
tff(fact_8533_ATP_Olambda__353,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,complex,aTP_Lamp_gu(fun(nat,real),fun(nat,complex),Uu),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_353
tff(fact_8534_ATP_Olambda__354,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_gy(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_354
tff(fact_8535_ATP_Olambda__355,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_gx(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_355
tff(fact_8536_ATP_Olambda__356,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_gw(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_356
tff(fact_8537_ATP_Olambda__357,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_gv(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_357
tff(fact_8538_ATP_Olambda__358,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_hf(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_358
tff(fact_8539_ATP_Olambda__359,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_yb(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_359
tff(fact_8540_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_aqt(fun(A,real),fun(A,complex),Uu),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_360
tff(fact_8541_ATP_Olambda__361,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_apc(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_361
tff(fact_8542_ATP_Olambda__362,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_aaa(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_362
tff(fact_8543_ATP_Olambda__363,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_adl(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_363
tff(fact_8544_ATP_Olambda__364,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_adm(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_364
tff(fact_8545_ATP_Olambda__365,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_365
tff(fact_8546_ATP_Olambda__366,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_366
tff(fact_8547_ATP_Olambda__367,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_367
tff(fact_8548_ATP_Olambda__368,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_sc(fun(B,set(A)),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_368
tff(fact_8549_ATP_Olambda__369,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_sp(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_369
tff(fact_8550_ATP_Olambda__370,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ci(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_370
tff(fact_8551_ATP_Olambda__371,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_xy(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_371
tff(fact_8552_ATP_Olambda__372,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aot(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_372
tff(fact_8553_ATP_Olambda__373,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wj(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_373
tff(fact_8554_ATP_Olambda__374,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abe(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_374
tff(fact_8555_ATP_Olambda__375,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abb(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_375
tff(fact_8556_ATP_Olambda__376,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ajn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,uminus_uminus(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_376
tff(fact_8557_ATP_Olambda__377,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_add(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_377
tff(fact_8558_ATP_Olambda__378,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_hu(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_378
tff(fact_8559_ATP_Olambda__379,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_he(fun(B,int),fun(B,A),Uu),Uua) = ring_1_of_int(A,aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_379
tff(fact_8560_ATP_Olambda__380,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_bx(fun(B,int),fun(B,A),Uu),Uua) = ring_1_of_int(A,aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_380
tff(fact_8561_ATP_Olambda__381,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ii(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_381
tff(fact_8562_ATP_Olambda__382,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_aqo(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,artanh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_382
tff(fact_8563_ATP_Olambda__383,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aic(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_383
tff(fact_8564_ATP_Olambda__384,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aei(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_384
tff(fact_8565_ATP_Olambda__385,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_aou(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arsinh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_385
tff(fact_8566_ATP_Olambda__386,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abu(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_386
tff(fact_8567_ATP_Olambda__387,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_387
tff(fact_8568_ATP_Olambda__388,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zo(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_388
tff(fact_8569_ATP_Olambda__389,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_api(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_389
tff(fact_8570_ATP_Olambda__390,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_390
tff(fact_8571_ATP_Olambda__391,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ads(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ).

% ATP.lambda_391
tff(fact_8572_ATP_Olambda__392,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_392
tff(fact_8573_ATP_Olambda__393,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aqm(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_393
tff(fact_8574_ATP_Olambda__394,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_aqi(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_394
tff(fact_8575_ATP_Olambda__395,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aib(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_395
tff(fact_8576_ATP_Olambda__396,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_acz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_396
tff(fact_8577_ATP_Olambda__397,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_397
tff(fact_8578_ATP_Olambda__398,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aql(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_398
tff(fact_8579_ATP_Olambda__399,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,C,aTP_Lamp_apr(fun(A,product_prod(B,C)),fun(A,C),Uu),Uua) = aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_399
tff(fact_8580_ATP_Olambda__400,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,C,aTP_Lamp_zu(fun(A,product_prod(B,C)),fun(A,C),Uu),Uua) = aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_400
tff(fact_8581_ATP_Olambda__401,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,C,aTP_Lamp_aed(fun(A,product_prod(B,C)),fun(A,C),Uu),Uua) = aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_401
tff(fact_8582_ATP_Olambda__402,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,B,aTP_Lamp_apq(fun(A,product_prod(B,C)),fun(A,B),Uu),Uua) = aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_402
tff(fact_8583_ATP_Olambda__403,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,B,aTP_Lamp_zv(fun(A,product_prod(B,C)),fun(A,B),Uu),Uua) = aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_403
tff(fact_8584_ATP_Olambda__404,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,B,aTP_Lamp_aec(fun(A,product_prod(B,C)),fun(A,B),Uu),Uua) = aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_404
tff(fact_8585_ATP_Olambda__405,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aqc(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_405
tff(fact_8586_ATP_Olambda__406,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_abg(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_406
tff(fact_8587_ATP_Olambda__407,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aeq(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_407
tff(fact_8588_ATP_Olambda__408,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_fe(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_408
tff(fact_8589_ATP_Olambda__409,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_bu(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_409
tff(fact_8590_ATP_Olambda__410,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ht(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_410
tff(fact_8591_ATP_Olambda__411,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_apx(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_411
tff(fact_8592_ATP_Olambda__412,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abt(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_412
tff(fact_8593_ATP_Olambda__413,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ael(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_413
tff(fact_8594_ATP_Olambda__414,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aqh(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_414
tff(fact_8595_ATP_Olambda__415,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_abk(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_415
tff(fact_8596_ATP_Olambda__416,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_xc(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tanh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_416
tff(fact_8597_ATP_Olambda__417,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ada(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_417
tff(fact_8598_ATP_Olambda__418,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aow(fun(A,B),fun(A,B),Uu),Uua) = sinh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_418
tff(fact_8599_ATP_Olambda__419,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_aaz(fun(A,B),fun(A,B),Uu),Uua) = sinh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_419
tff(fact_8600_ATP_Olambda__420,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_we(fun(A,A),fun(A,A),Uu),Uua) = sinh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_420
tff(fact_8601_ATP_Olambda__421,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_adh(fun(A,B),fun(A,B),Uu),Uua) = sinh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_421
tff(fact_8602_ATP_Olambda__422,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aox(fun(A,B),fun(A,B),Uu),Uua) = cosh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_422
tff(fact_8603_ATP_Olambda__423,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_aay(fun(A,B),fun(A,B),Uu),Uua) = cosh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_423
tff(fact_8604_ATP_Olambda__424,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wd(fun(A,A),fun(A,A),Uu),Uua) = cosh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_424
tff(fact_8605_ATP_Olambda__425,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_adi(fun(A,B),fun(A,B),Uu),Uua) = cosh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_425
tff(fact_8606_ATP_Olambda__426,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_426
tff(fact_8607_ATP_Olambda__427,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_abi(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_427
tff(fact_8608_ATP_Olambda__428,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yp(fun(A,real),fun(A,real),Uu),Uua) = sin(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_428
tff(fact_8609_ATP_Olambda__429,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_aaw(fun(A,B),fun(A,B),Uu),Uua) = sin(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_429
tff(fact_8610_ATP_Olambda__430,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_vu(fun(A,A),fun(A,A),Uu),Uua) = sin(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_430
tff(fact_8611_ATP_Olambda__431,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,exp(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_431
tff(fact_8612_ATP_Olambda__432,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_apb(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_432
tff(fact_8613_ATP_Olambda__433,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_aas(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_433
tff(fact_8614_ATP_Olambda__434,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_vv(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,exp(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_434
tff(fact_8615_ATP_Olambda__435,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ado(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_435
tff(fact_8616_ATP_Olambda__436,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_ia(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_436
tff(fact_8617_ATP_Olambda__437,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_abj(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_437
tff(fact_8618_ATP_Olambda__438,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ahf(fun(nat,real),fun(nat,real),Uu),Uua) = cos(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_438
tff(fact_8619_ATP_Olambda__439,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_yy(fun(A,real),fun(A,real),Uu),Uua) = cos(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_439
tff(fact_8620_ATP_Olambda__440,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_aax(fun(A,B),fun(A,B),Uu),Uua) = cos(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_440
tff(fact_8621_ATP_Olambda__441,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wp(fun(A,A),fun(A,A),Uu),Uua) = cos(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_441
tff(fact_8622_ATP_Olambda__442,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_bbe(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_442
tff(fact_8623_ATP_Olambda__443,axiom,
    ! [Uu: fun(nat,complex),Uua: nat] : aa(nat,real,aTP_Lamp_ou(fun(nat,complex),fun(nat,real),Uu),Uua) = re(aa(nat,complex,Uu,Uua)) ).

% ATP.lambda_443
tff(fact_8624_ATP_Olambda__444,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aaf(fun(A,complex),fun(A,real),Uu),Uua) = re(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_444
tff(fact_8625_ATP_Olambda__445,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aqx(fun(A,complex),fun(A,real),Uu),Uua) = re(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_445
tff(fact_8626_ATP_Olambda__446,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_abr(fun(A,complex),fun(A,real),Uu),Uua) = re(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_446
tff(fact_8627_ATP_Olambda__447,axiom,
    ! [A: $tType,Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ot(fun(A,complex),fun(A,real),Uu),Uua) = re(aa(A,complex,Uu,Uua)) ).

% ATP.lambda_447
tff(fact_8628_ATP_Olambda__448,axiom,
    ! [Uu: fun(nat,complex),Uua: nat] : aa(nat,real,aTP_Lamp_ow(fun(nat,complex),fun(nat,real),Uu),Uua) = im(aa(nat,complex,Uu,Uua)) ).

% ATP.lambda_448
tff(fact_8629_ATP_Olambda__449,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aae(fun(A,complex),fun(A,real),Uu),Uua) = im(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_449
tff(fact_8630_ATP_Olambda__450,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_aqv(fun(A,complex),fun(A,real),Uu),Uua) = im(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_450
tff(fact_8631_ATP_Olambda__451,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_abs(fun(A,complex),fun(A,real),Uu),Uua) = im(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_451
tff(fact_8632_ATP_Olambda__452,axiom,
    ! [A: $tType,Uu: fun(A,complex),Uua: A] : aa(A,real,aTP_Lamp_ov(fun(A,complex),fun(A,real),Uu),Uua) = im(aa(A,complex,Uu,Uua)) ).

% ATP.lambda_452
tff(fact_8633_ATP_Olambda__453,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(nat,set(A)),Uua: nat] : aa(nat,filter(A),aTP_Lamp_aoh(fun(nat,set(A)),fun(nat,filter(A)),Uu),Uua) = aa(set(A),filter(A),principal(A),aa(nat,set(A),Uu,Uua)) ) ).

% ATP.lambda_453
tff(fact_8634_ATP_Olambda__454,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_aiw(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = aa(set(C),filter(C),principal(C),aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_454
tff(fact_8635_ATP_Olambda__455,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,set(D)),Uua: C] : aa(C,filter(D),aTP_Lamp_aiv(fun(C,set(D)),fun(C,filter(D)),Uu),Uua) = aa(set(D),filter(D),principal(D),aa(C,set(D),Uu,Uua)) ).

% ATP.lambda_455
tff(fact_8636_ATP_Olambda__456,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,filter(A),aTP_Lamp_air(fun(B,set(A)),fun(B,filter(A)),Uu),Uua) = aa(set(A),filter(A),principal(A),aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_456
tff(fact_8637_ATP_Olambda__457,axiom,
    ! [E3: $tType,A: $tType,Uu: fun(A,set(E3)),Uua: A] : aa(A,filter(E3),aTP_Lamp_aiu(fun(A,set(E3)),fun(A,filter(E3)),Uu),Uua) = aa(set(E3),filter(E3),principal(E3),aa(A,set(E3),Uu,Uua)) ).

% ATP.lambda_457
tff(fact_8638_ATP_Olambda__458,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_aix(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = aa(set(B),filter(B),principal(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_458
tff(fact_8639_ATP_Olambda__459,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_su(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_459
tff(fact_8640_ATP_Olambda__460,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_xb(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_460
tff(fact_8641_ATP_Olambda__461,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_afj(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_461
tff(fact_8642_ATP_Olambda__462,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_fj(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_462
tff(fact_8643_ATP_Olambda__463,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_atz(fun(B,list(A)),fun(B,set(A)),Uu),Uua) = aa(list(A),set(A),set2(A),aa(B,list(A),Uu,Uua)) ).

% ATP.lambda_463
tff(fact_8644_ATP_Olambda__464,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zm(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_464
tff(fact_8645_ATP_Olambda__465,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aph(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_465
tff(fact_8646_ATP_Olambda__466,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_466
tff(fact_8647_ATP_Olambda__467,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_adp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ).

% ATP.lambda_467
tff(fact_8648_ATP_Olambda__468,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: A] : aa(A,set(B),aTP_Lamp_ava(fun(A,fun(B,$o)),fun(A,set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),Uu,Uua)) ).

% ATP.lambda_468
tff(fact_8649_ATP_Olambda__469,axiom,
    ! [Uu: fun(nat,complex),Uua: nat] : aa(nat,complex,aTP_Lamp_oz(fun(nat,complex),fun(nat,complex),Uu),Uua) = cnj(aa(nat,complex,Uu,Uua)) ).

% ATP.lambda_469
tff(fact_8650_ATP_Olambda__470,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_aad(fun(A,complex),fun(A,complex),Uu),Uua) = cnj(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_470
tff(fact_8651_ATP_Olambda__471,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_aqu(fun(A,complex),fun(A,complex),Uu),Uua) = cnj(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_471
tff(fact_8652_ATP_Olambda__472,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_aby(fun(A,complex),fun(A,complex),Uu),Uua) = cnj(aa(A,complex,Uu,Uua)) ) ).

% ATP.lambda_472
tff(fact_8653_ATP_Olambda__473,axiom,
    ! [A: $tType,Uu: fun(A,complex),Uua: A] : aa(A,complex,aTP_Lamp_oy(fun(A,complex),fun(A,complex),Uu),Uua) = cnj(aa(A,complex,Uu,Uua)) ).

% ATP.lambda_473
tff(fact_8654_ATP_Olambda__474,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_aqw(fun(A,real),fun(A,complex),Uu),Uua) = cis(aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_474
tff(fact_8655_ATP_Olambda__475,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_axi(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = insert(B,aa(A,B,Uu,Uua)) ).

% ATP.lambda_475
tff(fact_8656_ATP_Olambda__476,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_rr(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow2(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_476
tff(fact_8657_ATP_Olambda__477,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_pn(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_477
tff(fact_8658_ATP_Olambda__478,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_awe(fun(B,$o),fun(B,$o),Uu),Uua)
    <=> ~ aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_478
tff(fact_8659_ATP_Olambda__479,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aer(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ~ aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_479
tff(fact_8660_ATP_Olambda__480,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_fy(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_480
tff(fact_8661_ATP_Olambda__481,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ann(fun(B,fun(A,$o)),fun(B,$o),Uu),Uua)
        <=> ! [X_1: A] : aa(A,$o,aa(B,fun(A,$o),Uu,Uua),X_1) ) ) ).

% ATP.lambda_481
tff(fact_8662_ATP_Olambda__482,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_anu(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ! [X_1: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_1) ) ).

% ATP.lambda_482
tff(fact_8663_ATP_Olambda__483,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ayn(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ? [X_1: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_1) ) ).

% ATP.lambda_483
tff(fact_8664_ATP_Olambda__484,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real] : aa(real,filter(A),aTP_Lamp_aiz(A,fun(real,filter(A)),Uu),Uua) = aa(set(A),filter(A),principal(A),aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_aiy(A,fun(real,fun(A,$o)),Uu),Uua))) ) ).

% ATP.lambda_484
tff(fact_8665_ATP_Olambda__485,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(real,A),Uua: real] : aa(real,real,aTP_Lamp_abm(fun(real,A),fun(real,real),Uu),Uua) = ring_1_of_int(real,archim6421214686448440834_floor(A,aa(real,A,Uu,Uua))) ) ).

% ATP.lambda_485
tff(fact_8666_ATP_Olambda__486,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_adf(fun(A,B),fun(A,C),Uu),Uua) = ring_1_of_int(C,archim6421214686448440834_floor(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_486
tff(fact_8667_ATP_Olambda__487,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_adu(fun(A,B),fun(A,C),Uu),Uua) = ring_1_of_int(C,archimedean_ceiling(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_487
tff(fact_8668_ATP_Olambda__488,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o)] : aa(fun(B,$o),filter(product_prod(A,B)),aa(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B))),aTP_Lamp_ary(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))),Uu),Uua) = aa(set(product_prod(A,B)),filter(product_prod(A,B)),principal(product_prod(A,B)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_arx(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua)))) ).

% ATP.lambda_488
tff(fact_8669_ATP_Olambda__489,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,fun(B,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_asf(fun(A,fun(B,$o)),fun(B,$o),Uu),Uua)
        <=> ! [I4: A] : aa(B,$o,aa(A,fun(B,$o),Uu,I4),Uua) ) ) ).

% ATP.lambda_489
tff(fact_8670_ATP_Olambda__490,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,fun(B,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_ayt(fun(A,fun(B,$o)),fun(B,$o),Uu),Uua)
        <=> ? [I4: A] : aa(B,$o,aa(A,fun(B,$o),Uu,I4),Uua) ) ) ).

% ATP.lambda_490
tff(fact_8671_ATP_Olambda__491,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_azr(list(A),fun(A,$o),Uu),Uua)
    <=> ? [I4: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu)) ) ) ).

% ATP.lambda_491
tff(fact_8672_ATP_Olambda__492,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
      ( aa(set(B),$o,aTP_Lamp_azz(set(set(B)),fun(set(B),$o),Uu),Uua)
    <=> ? [F5: fun(set(B),B)] :
          ( ( Uua = aa(set(set(B)),set(B),image(set(B),B,F5),Uu) )
          & ! [X: set(B)] :
              ( aa(set(set(B)),$o,member(set(B),X),Uu)
             => aa(set(B),$o,member(B,aa(set(B),B,F5,X)),X) ) ) ) ).

% ATP.lambda_492
tff(fact_8673_ATP_Olambda__493,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_azv(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F5),Uu) )
              & ! [X: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),X),Uu)
                 => aa(set(A),$o,member(A,aa(set(A),A,F5,X)),X) ) ) ) ) ).

% ATP.lambda_493
tff(fact_8674_ATP_Olambda__494,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_azw(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F5),Uu) )
              & ! [X: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),X),Uu)
                 => aa(set(A),$o,member(A,aa(set(A),A,F5,X)),X) ) ) ) ) ).

% ATP.lambda_494
tff(fact_8675_ATP_Olambda__495,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_bak(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F5),Uu) )
              & ! [X: set(A)] :
                  ( aa(set(set(A)),$o,member(set(A),X),Uu)
                 => aa(set(A),$o,member(A,aa(set(A),A,F5,X)),X) ) ) ) ) ).

% ATP.lambda_495
tff(fact_8676_ATP_Olambda__496,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_bac(set(A),fun(set(A),$o),Uu),Uua)
    <=> ? [B10: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B10) )
          & aa(set(set(A)),$o,member(set(A),Uu),pow2(A,B10)) ) ) ).

% ATP.lambda_496
tff(fact_8677_ATP_Olambda__497,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
      ( aa(filter(A),$o,aTP_Lamp_baf(set(filter(A)),fun(filter(A),$o),Uu),Uua)
    <=> ! [X: filter(A)] :
          ( aa(set(filter(A)),$o,member(filter(A),X),Uu)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Uua),X) ) ) ).

% ATP.lambda_497
tff(fact_8678_ATP_Olambda__498,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_azs(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( aa(set(A),$o,member(A,X),Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X) ) ) ) ).

% ATP.lambda_498
tff(fact_8679_ATP_Olambda__499,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aze(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( aa(set(A),$o,member(A,X),Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X) ) ) ) ).

% ATP.lambda_499
tff(fact_8680_ATP_Olambda__500,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_azt(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( aa(set(A),$o,member(A,X),Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Uua) ) ) ) ).

% ATP.lambda_500
tff(fact_8681_ATP_Olambda__501,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_azf(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( aa(set(A),$o,member(A,X),Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Uua) ) ) ) ).

% ATP.lambda_501
tff(fact_8682_ATP_Olambda__502,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_baa(set(set(A)),fun(A,$o),Uu),Uua)
    <=> ! [X: set(A)] :
          ( aa(set(set(A)),$o,member(set(A),X),Uu)
         => aa(set(A),$o,member(A,Uua),X) ) ) ).

% ATP.lambda_502
tff(fact_8683_ATP_Olambda__503,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ant(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ! [Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y3)
             => aa(A,$o,Uu,Y3) ) ) ) ).

% ATP.lambda_503
tff(fact_8684_ATP_Olambda__504,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ayo(fun(B,A),fun(A,$o),Uu),Uua)
    <=> ? [X: B] : Uua = aa(B,A,Uu,X) ) ).

% ATP.lambda_504
tff(fact_8685_ATP_Olambda__505,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_axd(nat,fun(nat,set(nat)),Uu),Uua) = aa(nat,set(nat),set_ord_atMost(nat),Uu) ).

% ATP.lambda_505
tff(fact_8686_ATP_Olambda__506,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_auo(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).

% ATP.lambda_506
tff(fact_8687_ATP_Olambda__507,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_auq(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).

% ATP.lambda_507
tff(fact_8688_ATP_Olambda__508,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,set(B),aTP_Lamp_aul(fun(B,$o),fun(A,set(B)),Uu),Uua) = aa(fun(B,$o),set(B),collect(B),Uu) ).

% ATP.lambda_508
tff(fact_8689_ATP_Olambda__509,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_gl(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2)))),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),one_one(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% ATP.lambda_509
tff(fact_8690_ATP_Olambda__510,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_eb(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_510
tff(fact_8691_ATP_Olambda__511,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_pa(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),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),aa(num,num,bit0,one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(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),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_511
tff(fact_8692_ATP_Olambda__512,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bi(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),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),aa(num,num,bit0,one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(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),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_512
tff(fact_8693_ATP_Olambda__513,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_bj(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),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),aa(num,num,bit0,one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(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),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_513
tff(fact_8694_ATP_Olambda__514,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_bk(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),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),aa(num,num,bit0,one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(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),aa(num,num,bit0,one2))),Uua)),Uub)) ) ).

% ATP.lambda_514
tff(fact_8695_ATP_Olambda__515,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_eq(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(aa(set(nat),$o,member(nat,Uub),Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_515
tff(fact_8696_ATP_Olambda__516,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_uh(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(aa(set(A),$o,member(A,Uub),Uua),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_516
tff(fact_8697_ATP_Olambda__517,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_ui(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(aa(set(A),$o,member(A,Uub),Uua),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_517
tff(fact_8698_ATP_Olambda__518,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bv(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_518
tff(fact_8699_ATP_Olambda__519,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hh(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_519
tff(fact_8700_ATP_Olambda__520,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_em(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_520
tff(fact_8701_ATP_Olambda__521,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bw(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_521
tff(fact_8702_ATP_Olambda__522,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hg(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_522
tff(fact_8703_ATP_Olambda__523,axiom,
    ! [Uu: code_integer,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_qh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),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,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),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),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_523
tff(fact_8704_ATP_Olambda__524,axiom,
    ! [Uu: code_integer,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_vg(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),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,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),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),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).

% ATP.lambda_524
tff(fact_8705_ATP_Olambda__525,axiom,
    ! [Uu: code_integer,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_qg(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),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,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),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),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).

% ATP.lambda_525
tff(fact_8706_ATP_Olambda__526,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: set(A)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_axm(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),insert(A,Uua),Uub),Uub) ).

% ATP.lambda_526
tff(fact_8707_ATP_Olambda__527,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,$o),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_er(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(aa(nat,$o,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_527
tff(fact_8708_ATP_Olambda__528,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_cp(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_528
tff(fact_8709_ATP_Olambda__529,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_hv(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_529
tff(fact_8710_ATP_Olambda__530,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A,Uub: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(A,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_axk(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_axj(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_530
tff(fact_8711_ATP_Olambda__531,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B,Uub: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(B,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_ayb(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_axh(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_531
tff(fact_8712_ATP_Olambda__532,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( aa(real,$o,aa(fun(real,real),fun(real,$o),aTP_Lamp_akf(fun(real,real),fun(fun(real,real),fun(real,$o)),Uu),Uua),Uub)
    <=> has_field_derivative(real,Uu,aa(real,real,Uua,Uub),topolo174197925503356063within(real,Uub,top_top(set(real)))) ) ).

% ATP.lambda_532
tff(fact_8713_ATP_Olambda__533,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: filter(B),Uua: fun(B,A),Uub: A] :
          ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_ahy(filter(B),fun(fun(B,A),fun(A,$o)),Uu),Uua),Uub)
        <=> filterlim(B,A,Uua,topolo7230453075368039082e_nhds(A,Uub),Uu) ) ) ).

% ATP.lambda_533
tff(fact_8714_ATP_Olambda__534,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_atu(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uub),Uu),Uua) ).

% ATP.lambda_534
tff(fact_8715_ATP_Olambda__535,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: $o] :
      aa($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_aww(set(A),fun(set(A),fun($o,set(A))),Uu),Uua),(Uub)) = $ite((Uub),Uu,Uua) ).

% ATP.lambda_535
tff(fact_8716_ATP_Olambda__536,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_qe(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),Uu),Uua),Uub) = aa(B,C,aa(D,fun(B,C),Uu,aa(A,D,Uua,aa(product_prod(A,B),A,product_fst(A,B),Uub))),aa(product_prod(A,B),B,product_snd(A,B),Uub)) ).

% ATP.lambda_536
tff(fact_8717_ATP_Olambda__537,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_tw(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,aa(B,C,Uua,Uub)),Uub) ) ).

% ATP.lambda_537
tff(fact_8718_ATP_Olambda__538,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_jf(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,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_538
tff(fact_8719_ATP_Olambda__539,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_jd(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,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_539
tff(fact_8720_ATP_Olambda__540,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_azb(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_540
tff(fact_8721_ATP_Olambda__541,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_xa(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_541
tff(fact_8722_ATP_Olambda__542,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_ix(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_542
tff(fact_8723_ATP_Olambda__543,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_iu(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_543
tff(fact_8724_ATP_Olambda__544,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_tu(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_544
tff(fact_8725_ATP_Olambda__545,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Sup(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_sa(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_545
tff(fact_8726_ATP_Olambda__546,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( complete_Inf(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_rf(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).

% ATP.lambda_546
tff(fact_8727_ATP_Olambda__547,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_vx(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_547
tff(fact_8728_ATP_Olambda__548,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_rm(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_548
tff(fact_8729_ATP_Olambda__549,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_hj(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_549
tff(fact_8730_ATP_Olambda__550,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_bz(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_550
tff(fact_8731_ATP_Olambda__551,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_anm(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ) ).

% ATP.lambda_551
tff(fact_8732_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ayk(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_552
tff(fact_8733_ATP_Olambda__553,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_aej(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_553
tff(fact_8734_ATP_Olambda__554,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_aee(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_554
tff(fact_8735_ATP_Olambda__555,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_avc(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_555
tff(fact_8736_ATP_Olambda__556,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_fc(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(A,fun(nat,B),Uu,Uub),Uua) ) ).

% ATP.lambda_556
tff(fact_8737_ATP_Olambda__557,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_yh(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_557
tff(fact_8738_ATP_Olambda__558,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_zg(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_558
tff(fact_8739_ATP_Olambda__559,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_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_apy(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_559
tff(fact_8740_ATP_Olambda__560,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_aqa(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_560
tff(fact_8741_ATP_Olambda__561,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_apu(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_561
tff(fact_8742_ATP_Olambda__562,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_aag(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_562
tff(fact_8743_ATP_Olambda__563,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_aai(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_563
tff(fact_8744_ATP_Olambda__564,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_aak(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_564
tff(fact_8745_ATP_Olambda__565,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_anv(fun(A,fun(B,$o)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_565
tff(fact_8746_ATP_Olambda__566,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_aeo(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_566
tff(fact_8747_ATP_Olambda__567,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_aem(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_567
tff(fact_8748_ATP_Olambda__568,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_aeg(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_568
tff(fact_8749_ATP_Olambda__569,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_hr(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_569
tff(fact_8750_ATP_Olambda__570,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_cl(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_570
tff(fact_8751_ATP_Olambda__571,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_kp(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_ko(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_571
tff(fact_8752_ATP_Olambda__572,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_kn(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_km(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_572
tff(fact_8753_ATP_Olambda__573,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_kl(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_kk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_573
tff(fact_8754_ATP_Olambda__574,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_jo(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_jn(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_574
tff(fact_8755_ATP_Olambda__575,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_gf(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu,one_one(A),zero_zero(A))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_575
tff(fact_8756_ATP_Olambda__576,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_qb(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_576
tff(fact_8757_ATP_Olambda__577,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_iy(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ix(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_577
tff(fact_8758_ATP_Olambda__578,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_iv(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_iu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_578
tff(fact_8759_ATP_Olambda__579,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_abl(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_fh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_579
tff(fact_8760_ATP_Olambda__580,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_vp($o,fun(code_integer,fun($o,char)),(Uu)),Uua),(Uub)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_vo($o,fun($o,fun(code_integer,fun($o,char))),(Uu)),(Uub))),code_bit_cut_integer(Uua)) ).

% ATP.lambda_580
tff(fact_8761_ATP_Olambda__581,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_nz(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(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_ny(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_581
tff(fact_8762_ATP_Olambda__582,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_nx(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(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_nw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_582
tff(fact_8763_ATP_Olambda__583,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_nv(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(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_nu(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_583
tff(fact_8764_ATP_Olambda__584,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_nt(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(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_ns(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_584
tff(fact_8765_ATP_Olambda__585,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_nm(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_nl(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_585
tff(fact_8766_ATP_Olambda__586,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_nj(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_ni(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_586
tff(fact_8767_ATP_Olambda__587,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_aqb(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_aqa(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_587
tff(fact_8768_ATP_Olambda__588,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(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_apz(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_apy(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_588
tff(fact_8769_ATP_Olambda__589,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_aep(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_aeo(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_589
tff(fact_8770_ATP_Olambda__590,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_aen(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_aem(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_590
tff(fact_8771_ATP_Olambda__591,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_aaj(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_aai(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_591
tff(fact_8772_ATP_Olambda__592,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_aah(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_aag(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_592
tff(fact_8773_ATP_Olambda__593,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(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_zh(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_zg(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_593
tff(fact_8774_ATP_Olambda__594,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_hk(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(C,fun(B,A),aTP_Lamp_hj(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_594
tff(fact_8775_ATP_Olambda__595,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_apv(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_apu(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_595
tff(fact_8776_ATP_Olambda__596,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_aeh(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_aeg(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_596
tff(fact_8777_ATP_Olambda__597,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_aal(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_aak(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_597
tff(fact_8778_ATP_Olambda__598,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_yi(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_yh(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_598
tff(fact_8779_ATP_Olambda__599,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(nat,B)),Uub: nat] : aa(nat,B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_fd(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),Uu),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(nat,fun(A,B),aTP_Lamp_fc(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_599
tff(fact_8780_ATP_Olambda__600,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_ca(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_bz(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_600
tff(fact_8781_ATP_Olambda__601,axiom,
    ! [D: $tType,E3: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(B,fun(C,fun(D,fun(E3,set(A))))),Uua: product_prod(B,C),Uub: product_prod(D,E3)] : aa(product_prod(D,E3),set(A),aa(product_prod(B,C),fun(product_prod(D,E3),set(A)),aTP_Lamp_sy(fun(B,fun(C,fun(D,fun(E3,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E3),set(A))),Uu),Uua),Uub) = aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(product_prod(D,E3),fun(B,fun(C,set(A))),aTP_Lamp_sx(fun(B,fun(C,fun(D,fun(E3,set(A))))),fun(product_prod(D,E3),fun(B,fun(C,set(A)))),Uu),Uub)),Uua) ).

% ATP.lambda_601
tff(fact_8782_ATP_Olambda__602,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_xg(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_602
tff(fact_8783_ATP_Olambda__603,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_xh(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_603
tff(fact_8784_ATP_Olambda__604,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_fu(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_604
tff(fact_8785_ATP_Olambda__605,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_jj(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_is(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_605
tff(fact_8786_ATP_Olambda__606,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_aff(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,aa(A,fun(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_606
tff(fact_8787_ATP_Olambda__607,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_afc(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,aa(A,fun(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_607
tff(fact_8788_ATP_Olambda__608,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_ct(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_608
tff(fact_8789_ATP_Olambda__609,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_aet(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,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_609
tff(fact_8790_ATP_Olambda__610,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_afd(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,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_610
tff(fact_8791_ATP_Olambda__611,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_xd(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_611
tff(fact_8792_ATP_Olambda__612,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_fv(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_612
tff(fact_8793_ATP_Olambda__613,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_gp(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_613
tff(fact_8794_ATP_Olambda__614,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_bav(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & aa(set(product_prod(list(A),list(A))),$o,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub)),lex(A,Uu)) ) ) ) ).

% ATP.lambda_614
tff(fact_8795_ATP_Olambda__615,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_bar(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys: list(A),X: A,Y3: A,Xs6: list(A),Ys7: list(A)] :
            ( ( Uua = append(A,Xys,cons(A,X,Xs6)) )
            & ( Uub = append(A,Xys,cons(A,Y3,Ys7)) )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),Uu) ) ) ) ).

% ATP.lambda_615
tff(fact_8796_ATP_Olambda__616,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_pv(nat,fun(set(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_616
tff(fact_8797_ATP_Olambda__617,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_pu(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_617
tff(fact_8798_ATP_Olambda__618,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_auj(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_618
tff(fact_8799_ATP_Olambda__619,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_ox(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_619
tff(fact_8800_ATP_Olambda__620,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ak(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_620
tff(fact_8801_ATP_Olambda__621,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_aj(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),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_621
tff(fact_8802_ATP_Olambda__622,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_wx(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_622
tff(fact_8803_ATP_Olambda__623,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_pk(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,aa(nat,nat,suc,Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_623
tff(fact_8804_ATP_Olambda__624,axiom,
    ! [A: $tType,Uu: set(list(A)),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_arb(set(list(A)),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(list(A)),$o,member(list(A),append(A,Uua,cons(A,Uub,nil(A)))),Uu) ) ).

% ATP.lambda_624
tff(fact_8805_ATP_Olambda__625,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_qa(nat,fun(nat,fun(set(nat),$o)),Uu),Uua),Uub)
    <=> ( aa(set(set(nat)),$o,member(set(nat),Uub),pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu)))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_625
tff(fact_8806_ATP_Olambda__626,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_on(A,fun(nat,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),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_626
tff(fact_8807_ATP_Olambda__627,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_pf(set(A),fun(nat,fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu)
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_627
tff(fact_8808_ATP_Olambda__628,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_pl(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(set(nat),$o,member(nat,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_628
tff(fact_8809_ATP_Olambda__629,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_go(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_629
tff(fact_8810_ATP_Olambda__630,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_gs(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_630
tff(fact_8811_ATP_Olambda__631,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_gt(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_631
tff(fact_8812_ATP_Olambda__632,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_jh(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub)) ).

% ATP.lambda_632
tff(fact_8813_ATP_Olambda__633,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_awj(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        | aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_633
tff(fact_8814_ATP_Olambda__634,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_eu(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_634
tff(fact_8815_ATP_Olambda__635,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_bp(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_635
tff(fact_8816_ATP_Olambda__636,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ar(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_636
tff(fact_8817_ATP_Olambda__637,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_bq(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_637
tff(fact_8818_ATP_Olambda__638,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_at(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_638
tff(fact_8819_ATP_Olambda__639,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_au(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_639
tff(fact_8820_ATP_Olambda__640,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_as(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_640
tff(fact_8821_ATP_Olambda__641,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ab(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uub) ) ) ).

% ATP.lambda_641
tff(fact_8822_ATP_Olambda__642,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aa(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_642
tff(fact_8823_ATP_Olambda__643,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_qj(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uu) ) ) ).

% ATP.lambda_643
tff(fact_8824_ATP_Olambda__644,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_qi(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uub),Uua)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Uub),Uu) ) ) ).

% ATP.lambda_644
tff(fact_8825_ATP_Olambda__645,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(set(A)),Uua: A,Uub: set(A)] :
          ( aa(set(A),$o,aa(A,fun(set(A),$o),aTP_Lamp_aol(set(set(A)),fun(A,fun(set(A),$o)),Uu),Uua),Uub)
        <=> ( aa(set(set(A)),$o,member(set(A),Uub),Uu)
            & aa(set(A),$o,member(A,Uua),Uub) ) ) ) ).

% ATP.lambda_645
tff(fact_8826_ATP_Olambda__646,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_tm(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_646
tff(fact_8827_ATP_Olambda__647,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: A] :
          ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_aqj(set(A),fun(fun(A,real),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_647
tff(fact_8828_ATP_Olambda__648,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ala(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uua)
       => aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_648
tff(fact_8829_ATP_Olambda__649,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_co(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_649
tff(fact_8830_ATP_Olambda__650,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_axo(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uua)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_650
tff(fact_8831_ATP_Olambda__651,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_fg(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_651
tff(fact_8832_ATP_Olambda__652,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_awa(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_652
tff(fact_8833_ATP_Olambda__653,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ff(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_653
tff(fact_8834_ATP_Olambda__654,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
      ( aa(A,$o,aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_auw(set(A),fun(fun(A,set(B)),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_654
tff(fact_8835_ATP_Olambda__655,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_az(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_655
tff(fact_8836_ATP_Olambda__656,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_oa(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_656
tff(fact_8837_ATP_Olambda__657,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ax(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_657
tff(fact_8838_ATP_Olambda__658,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ob(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_658
tff(fact_8839_ATP_Olambda__659,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_aus(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(set(B),$o,member(B,Uub),Uua)
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_659
tff(fact_8840_ATP_Olambda__660,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_pb(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),Uu)
            & ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2))),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_660
tff(fact_8841_ATP_Olambda__661,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_af(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,member(A,Uub),Uu)
        & ~ aa(set(A),$o,member(A,Uub),Uua) ) ) ).

% ATP.lambda_661
tff(fact_8842_ATP_Olambda__662,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ne(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_662
tff(fact_8843_ATP_Olambda__663,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jc(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_663
tff(fact_8844_ATP_Olambda__664,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aie(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_664
tff(fact_8845_ATP_Olambda__665,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_ark(real,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(complex,Uua,Uub)),Uu) ) ).

% ATP.lambda_665
tff(fact_8846_ATP_Olambda__666,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_ari(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(real,Uua,Uub)),Uu) ) ).

% ATP.lambda_666
tff(fact_8847_ATP_Olambda__667,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_arc(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_667
tff(fact_8848_ATP_Olambda__668,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ain(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_668
tff(fact_8849_ATP_Olambda__669,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aiy(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uub,Uu)),Uua) ) ) ).

% ATP.lambda_669
tff(fact_8850_ATP_Olambda__670,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_jl(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_670
tff(fact_8851_ATP_Olambda__671,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_nq(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_671
tff(fact_8852_ATP_Olambda__672,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_no(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_672
tff(fact_8853_ATP_Olambda__673,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_np(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_673
tff(fact_8854_ATP_Olambda__674,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_nn(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_674
tff(fact_8855_ATP_Olambda__675,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_qm(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu),Uua),Uub)
    <=> aa(set(product_prod(A,B)),$o,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_675
tff(fact_8856_ATP_Olambda__676,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_avz(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu) ) ).

% ATP.lambda_676
tff(fact_8857_ATP_Olambda__677,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_aw(nat,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu) = Uua ) ) ).

% ATP.lambda_677
tff(fact_8858_ATP_Olambda__678,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_lk(complex,fun(nat,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu ) ) ).

% ATP.lambda_678
tff(fact_8859_ATP_Olambda__679,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_mg(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(A),$o,member(A,Uub),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_679
tff(fact_8860_ATP_Olambda__680,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,A),Uub: A] :
      ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_atp(fun(A,B),fun(fun(B,A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uua,aa(A,B,Uu,Uub)) = Uub ) ) ).

% ATP.lambda_680
tff(fact_8861_ATP_Olambda__681,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fm(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_681
tff(fact_8862_ATP_Olambda__682,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_fo(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_682
tff(fact_8863_ATP_Olambda__683,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(B,$o),fun(product_prod(A,B),$o),aTP_Lamp_arp(fun(A,$o),fun(fun(B,$o),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uub))
        & aa(B,$o,Uua,aa(product_prod(A,B),B,product_snd(A,B),Uub)) ) ) ).

% ATP.lambda_683
tff(fact_8864_ATP_Olambda__684,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ahe(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),aa(real,real,aa(real,fun(real,real),times_times(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),aa(num,num,bit0,one2))),pi))) ).

% ATP.lambda_684
tff(fact_8865_ATP_Olambda__685,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_xe(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_685
tff(fact_8866_ATP_Olambda__686,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_fs(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_686
tff(fact_8867_ATP_Olambda__687,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ju(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_687
tff(fact_8868_ATP_Olambda__688,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_abn(fun(nat,B),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uub)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),Uub)) ) ).

% ATP.lambda_688
tff(fact_8869_ATP_Olambda__689,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_is(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_689
tff(fact_8870_ATP_Olambda__690,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fh(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_690
tff(fact_8871_ATP_Olambda__691,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_fn(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_691
tff(fact_8872_ATP_Olambda__692,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_gr(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_692
tff(fact_8873_ATP_Olambda__693,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_jb(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_693
tff(fact_8874_ATP_Olambda__694,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_jk(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_694
tff(fact_8875_ATP_Olambda__695,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ah(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_695
tff(fact_8876_ATP_Olambda__696,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_xu(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_696
tff(fact_8877_ATP_Olambda__697,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(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_aoy(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_697
tff(fact_8878_ATP_Olambda__698,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(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_aab(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_698
tff(fact_8879_ATP_Olambda__699,axiom,
    ! [B: $tType,A: $tType] :
      ( 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_adj(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_699
tff(fact_8880_ATP_Olambda__700,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_avr(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uub)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_700
tff(fact_8881_ATP_Olambda__701,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_avq(fun(A,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)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_701
tff(fact_8882_ATP_Olambda__702,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_aop(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_702
tff(fact_8883_ATP_Olambda__703,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_aif(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_703
tff(fact_8884_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_aih(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_704
tff(fact_8885_ATP_Olambda__705,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_asi(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_705
tff(fact_8886_ATP_Olambda__706,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] :
      ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_ama(fun(A,real),fun(fun(A,real),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_706
tff(fact_8887_ATP_Olambda__707,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_alm(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_707
tff(fact_8888_ATP_Olambda__708,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_als(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_708
tff(fact_8889_ATP_Olambda__709,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_amo(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_709
tff(fact_8890_ATP_Olambda__710,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_anj(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_710
tff(fact_8891_ATP_Olambda__711,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_ho(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_711
tff(fact_8892_ATP_Olambda__712,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ze(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_712
tff(fact_8893_ATP_Olambda__713,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_yr(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_713
tff(fact_8894_ATP_Olambda__714,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_aps(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_714
tff(fact_8895_ATP_Olambda__715,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_wm(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_715
tff(fact_8896_ATP_Olambda__716,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_abc(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_716
tff(fact_8897_ATP_Olambda__717,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_ajp(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_717
tff(fact_8898_ATP_Olambda__718,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajw(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_718
tff(fact_8899_ATP_Olambda__719,axiom,
    ! [B: $tType,A: $tType] :
      ( 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_acq(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_719
tff(fact_8900_ATP_Olambda__720,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_akg(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_720
tff(fact_8901_ATP_Olambda__721,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aqe(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_721
tff(fact_8902_ATP_Olambda__722,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_or(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_722
tff(fact_8903_ATP_Olambda__723,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_ans(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_723
tff(fact_8904_ATP_Olambda__724,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_oq(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_724
tff(fact_8905_ATP_Olambda__725,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_hn(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_725
tff(fact_8906_ATP_Olambda__726,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_yj(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_726
tff(fact_8907_ATP_Olambda__727,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_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_apm(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_727
tff(fact_8908_ATP_Olambda__728,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_apl(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_728
tff(fact_8909_ATP_Olambda__729,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_wc(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_729
tff(fact_8910_ATP_Olambda__730,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_aao(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_730
tff(fact_8911_ATP_Olambda__731,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aap(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_731
tff(fact_8912_ATP_Olambda__732,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_ajf(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_732
tff(fact_8913_ATP_Olambda__733,axiom,
    ! [B: $tType,A: $tType] :
      ( 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_acy(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_733
tff(fact_8914_ATP_Olambda__734,axiom,
    ! [B: $tType,A: $tType] :
      ( 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_aju(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_734
tff(fact_8915_ATP_Olambda__735,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adv(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_735
tff(fact_8916_ATP_Olambda__736,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ady(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_736
tff(fact_8917_ATP_Olambda__737,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aur(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_737
tff(fact_8918_ATP_Olambda__738,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_ajs(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_738
tff(fact_8919_ATP_Olambda__739,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_cr(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_739
tff(fact_8920_ATP_Olambda__740,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_agf(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_740
tff(fact_8921_ATP_Olambda__741,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_ez(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_741
tff(fact_8922_ATP_Olambda__742,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_cg(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_742
tff(fact_8923_ATP_Olambda__743,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_yg(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_743
tff(fact_8924_ATP_Olambda__744,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_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_apo(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_744
tff(fact_8925_ATP_Olambda__745,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_vz(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_745
tff(fact_8926_ATP_Olambda__746,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_abd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_746
tff(fact_8927_ATP_Olambda__747,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_aam(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_747
tff(fact_8928_ATP_Olambda__748,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_aut(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_748
tff(fact_8929_ATP_Olambda__749,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_acs(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_749
tff(fact_8930_ATP_Olambda__750,axiom,
    ! [B: $tType,A: $tType] :
      ( 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_aea(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_750
tff(fact_8931_ATP_Olambda__751,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_gg(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_751
tff(fact_8932_ATP_Olambda__752,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_dt(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_752
tff(fact_8933_ATP_Olambda__753,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_cu(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_753
tff(fact_8934_ATP_Olambda__754,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_acu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_754
tff(fact_8935_ATP_Olambda__755,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_apf(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_755
tff(fact_8936_ATP_Olambda__756,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_aat(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_756
tff(fact_8937_ATP_Olambda__757,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_adq(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_757
tff(fact_8938_ATP_Olambda__758,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_aov(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_758
tff(fact_8939_ATP_Olambda__759,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_aba(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_759
tff(fact_8940_ATP_Olambda__760,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adg(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_760
tff(fact_8941_ATP_Olambda__761,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_adt(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_761
tff(fact_8942_ATP_Olambda__762,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_awt(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_762
tff(fact_8943_ATP_Olambda__763,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_awq(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_763
tff(fact_8944_ATP_Olambda__764,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_awf(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_764
tff(fact_8945_ATP_Olambda__765,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_awi(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_765
tff(fact_8946_ATP_Olambda__766,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_awv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_766
tff(fact_8947_ATP_Olambda__767,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_uc(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_767
tff(fact_8948_ATP_Olambda__768,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tt(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_768
tff(fact_8949_ATP_Olambda__769,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aoo(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_769
tff(fact_8950_ATP_Olambda__770,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_auv(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_770
tff(fact_8951_ATP_Olambda__771,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aqn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_771
tff(fact_8952_ATP_Olambda__772,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_ey(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_772
tff(fact_8953_ATP_Olambda__773,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_cf(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_773
tff(fact_8954_ATP_Olambda__774,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_yf(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_774
tff(fact_8955_ATP_Olambda__775,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_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_apn(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_775
tff(fact_8956_ATP_Olambda__776,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_wb(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_776
tff(fact_8957_ATP_Olambda__777,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_aan(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_777
tff(fact_8958_ATP_Olambda__778,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_ajm(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_778
tff(fact_8959_ATP_Olambda__779,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_ajo(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_779
tff(fact_8960_ATP_Olambda__780,axiom,
    ! [B: $tType,A: $tType] :
      ( 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_acv(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_780
tff(fact_8961_ATP_Olambda__781,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oc(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_781
tff(fact_8962_ATP_Olambda__782,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_wz(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_782
tff(fact_8963_ATP_Olambda__783,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_zk(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_783
tff(fact_8964_ATP_Olambda__784,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_aos(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_784
tff(fact_8965_ATP_Olambda__785,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_abz(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_785
tff(fact_8966_ATP_Olambda__786,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_adc(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_786
tff(fact_8967_ATP_Olambda__787,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_aqk(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_787
tff(fact_8968_ATP_Olambda__788,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_acb(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_788
tff(fact_8969_ATP_Olambda__789,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_aco(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_789
tff(fact_8970_ATP_Olambda__790,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,product_prod(B,B),aa(A,fun(A,product_prod(B,B)),aTP_Lamp_avs(fun(A,B),fun(A,fun(A,product_prod(B,B))),Uu),Uua),Uub) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uu,Uua)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_790
tff(fact_8971_ATP_Olambda__791,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_atj(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).

% ATP.lambda_791
tff(fact_8972_ATP_Olambda__792,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_app(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_792
tff(fact_8973_ATP_Olambda__793,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_zw(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_793
tff(fact_8974_ATP_Olambda__794,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_aeb(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_794
tff(fact_8975_ATP_Olambda__795,axiom,
    ! [B: $tType,C: $tType,A: $tType,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_ars(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_795
tff(fact_8976_ATP_Olambda__796,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_asd(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,$o,Uu,Uub)
           => aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_796
tff(fact_8977_ATP_Olambda__797,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_alf(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_797
tff(fact_8978_ATP_Olambda__798,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,B)),Uua: fun(A,nat),Uub: A] : aa(A,fun(B,B),aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_axz(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),Uu),Uua),Uub) = aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(A,nat,Uua,Uub)),aa(A,fun(B,B),Uu,Uub)) ).

% ATP.lambda_798
tff(fact_8979_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahw(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,$o,Uu,Uub)
            | aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_799
tff(fact_8980_ATP_Olambda__800,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aq(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_800
tff(fact_8981_ATP_Olambda__801,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_vh(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_801
tff(fact_8982_ATP_Olambda__802,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ahv(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,$o,Uu,Uub)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_802
tff(fact_8983_ATP_Olambda__803,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_803
tff(fact_8984_ATP_Olambda__804,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_akn(fun(nat,A),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> ( aa(nat,A,Uu,Uub) = aa(nat,A,Uua,Uub) ) ) ) ).

% ATP.lambda_804
tff(fact_8985_ATP_Olambda__805,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_akt(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_805
tff(fact_8986_ATP_Olambda__806,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_all(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_806
tff(fact_8987_ATP_Olambda__807,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_asg(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_807
tff(fact_8988_ATP_Olambda__808,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aTP_Lamp_alk(fun(A,A),fun(fun(A,A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,A,Uu,Uub) = aa(A,A,Uua,Uub) ) ) ) ).

% ATP.lambda_808
tff(fact_8989_ATP_Olambda__809,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_amk(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_809
tff(fact_8990_ATP_Olambda__810,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_alg(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
      <=> aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_810
tff(fact_8991_ATP_Olambda__811,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aky(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_811
tff(fact_8992_ATP_Olambda__812,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_akq(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_812
tff(fact_8993_ATP_Olambda__813,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_tg(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_813
tff(fact_8994_ATP_Olambda__814,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ani(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),ring_1_of_int(B,archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_814
tff(fact_8995_ATP_Olambda__815,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aqs(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & ! [Y3: A] :
            ( aa(A,$o,Uua,Y3)
           => aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Y3) ) ) ) ).

% ATP.lambda_815
tff(fact_8996_ATP_Olambda__816,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_wr(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_816
tff(fact_8997_ATP_Olambda__817,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_apa(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = comm_s3205402744901411588hammer(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_817
tff(fact_8998_ATP_Olambda__818,axiom,
    ! [A: $tType,B: $tType] :
      ( order(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,set(B),aa(B,fun(A,set(B)),aTP_Lamp_avn(fun(A,B),fun(B,fun(A,set(B))),Uu),Uua),Uub) = set_or3652927894154168847AtMost(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_818
tff(fact_8999_ATP_Olambda__819,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,real),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ku(fun(nat,real),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_819
tff(fact_9000_ATP_Olambda__820,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(B,real),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_kt(fun(B,real),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(B,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_820
tff(fact_9001_ATP_Olambda__821,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,real),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ya(fun(A,real),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_821
tff(fact_9002_ATP_Olambda__822,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_hi(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_822
tff(fact_9003_ATP_Olambda__823,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_aek(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_823
tff(fact_9004_ATP_Olambda__824,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_by(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_824
tff(fact_9005_ATP_Olambda__825,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_vy(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_825
tff(fact_9006_ATP_Olambda__826,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_aef(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_826
tff(fact_9007_ATP_Olambda__827,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_aij(fun(A,B),fun(B,fun(A,real)),Uu),Uua),Uub) = real_V557655796197034286t_dist(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_827
tff(fact_9008_ATP_Olambda__828,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( aa(nat,$o,aa(set(A),fun(nat,$o),aTP_Lamp_amb(fun(nat,set(A)),fun(set(A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),Uu,Uub)),Uua) ) ) ).

% ATP.lambda_828
tff(fact_9009_ATP_Olambda__829,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ai(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_829
tff(fact_9010_ATP_Olambda__830,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_sv(fun(B,set(A)),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua) ) ).

% ATP.lambda_830
tff(fact_9011_ATP_Olambda__831,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_amm(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_831
tff(fact_9012_ATP_Olambda__832,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aml(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_832
tff(fact_9013_ATP_Olambda__833,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_amd(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_833
tff(fact_9014_ATP_Olambda__834,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aly(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_834
tff(fact_9015_ATP_Olambda__835,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_cn(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_835
tff(fact_9016_ATP_Olambda__836,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_ep(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_836
tff(fact_9017_ATP_Olambda__837,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_ch(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_837
tff(fact_9018_ATP_Olambda__838,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_wn(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_838
tff(fact_9019_ATP_Olambda__839,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_un(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_839
tff(fact_9020_ATP_Olambda__840,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_acp(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_840
tff(fact_9021_ATP_Olambda__841,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_afo(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_841
tff(fact_9022_ATP_Olambda__842,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_alq(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_842
tff(fact_9023_ATP_Olambda__843,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aln(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_843
tff(fact_9024_ATP_Olambda__844,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_alz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_844
tff(fact_9025_ATP_Olambda__845,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ew(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_845
tff(fact_9026_ATP_Olambda__846,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_cc(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_846
tff(fact_9027_ATP_Olambda__847,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ye(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_847
tff(fact_9028_ATP_Olambda__848,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_acj(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_848
tff(fact_9029_ATP_Olambda__849,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_apj(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_849
tff(fact_9030_ATP_Olambda__850,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_wf(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_850
tff(fact_9031_ATP_Olambda__851,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aar(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_851
tff(fact_9032_ATP_Olambda__852,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_acx(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_852
tff(fact_9033_ATP_Olambda__853,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_adx(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_853
tff(fact_9034_ATP_Olambda__854,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_yx(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_854
tff(fact_9035_ATP_Olambda__855,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_gb(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_855
tff(fact_9036_ATP_Olambda__856,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_afm(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_856
tff(fact_9037_ATP_Olambda__857,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_acg(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_857
tff(fact_9038_ATP_Olambda__858,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dv(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_858
tff(fact_9039_ATP_Olambda__859,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_sl(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_859
tff(fact_9040_ATP_Olambda__860,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_act(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_860
tff(fact_9041_ATP_Olambda__861,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_wv(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_861
tff(fact_9042_ATP_Olambda__862,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: nat,Uub: B] : aa(B,A,aa(nat,fun(B,A),aTP_Lamp_hp(fun(B,A),fun(nat,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_862
tff(fact_9043_ATP_Olambda__863,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_za(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_863
tff(fact_9044_ATP_Olambda__864,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_ape(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_864
tff(fact_9045_ATP_Olambda__865,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_ws(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_865
tff(fact_9046_ATP_Olambda__866,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_aav(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_866
tff(fact_9047_ATP_Olambda__867,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_ajx(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_867
tff(fact_9048_ATP_Olambda__868,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_acr(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_868
tff(fact_9049_ATP_Olambda__869,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_adr(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_869
tff(fact_9050_ATP_Olambda__870,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ajj(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_870
tff(fact_9051_ATP_Olambda__871,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aci(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_871
tff(fact_9052_ATP_Olambda__872,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_awh(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_872
tff(fact_9053_ATP_Olambda__873,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_awl(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_873
tff(fact_9054_ATP_Olambda__874,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ti(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_874
tff(fact_9055_ATP_Olambda__875,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ts(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_875
tff(fact_9056_ATP_Olambda__876,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_uf(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_876
tff(fact_9057_ATP_Olambda__877,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_anr(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_877
tff(fact_9058_ATP_Olambda__878,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_mq(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_878
tff(fact_9059_ATP_Olambda__879,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_wy(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_879
tff(fact_9060_ATP_Olambda__880,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ajt(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_880
tff(fact_9061_ATP_Olambda__881,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,filter(B)),Uua: filter(C),Uub: A] : aa(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_asc(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,aa(A,filter(B),Uu,Uub),Uua) ).

% ATP.lambda_881
tff(fact_9062_ATP_Olambda__882,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,set(product_prod(A,C))),Uua: set(product_prod(C,B)),Uub: D] : aa(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_axr(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,aa(D,set(product_prod(A,C)),Uu,Uub),Uua) ).

% ATP.lambda_882
tff(fact_9063_ATP_Olambda__883,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: int,Uub: B] : aa(B,A,aa(int,fun(B,A),aTP_Lamp_auc(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_883
tff(fact_9064_ATP_Olambda__884,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_auh(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_884
tff(fact_9065_ATP_Olambda__885,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_aue(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_885
tff(fact_9066_ATP_Olambda__886,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_aui(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_886
tff(fact_9067_ATP_Olambda__887,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_aug(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_887
tff(fact_9068_ATP_Olambda__888,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_auf(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_888
tff(fact_9069_ATP_Olambda__889,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_alx(fun(B,A),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
        <=> aa(set(A),$o,member(A,aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_889
tff(fact_9070_ATP_Olambda__890,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: set(B),Uub: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_alw(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(set(B),$o,member(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_890
tff(fact_9071_ATP_Olambda__891,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,member(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_891
tff(fact_9072_ATP_Olambda__892,axiom,
    ! [C: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(C,A),Uub: C] :
          ( aa(C,$o,aa(fun(C,A),fun(C,$o),aTP_Lamp_aqf(set(A),fun(fun(C,A),fun(C,$o)),Uu),Uua),Uub)
        <=> aa(set(A),$o,member(A,aa(C,A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_892
tff(fact_9073_ATP_Olambda__893,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(B,A),Uub: B] :
          ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_asl(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
        <=> aa(set(A),$o,member(A,aa(B,A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_893
tff(fact_9074_ATP_Olambda__894,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_asx(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_894
tff(fact_9075_ATP_Olambda__895,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_alh(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | (Uua) ) ) ).

% ATP.lambda_895
tff(fact_9076_ATP_Olambda__896,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akv(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_896
tff(fact_9077_ATP_Olambda__897,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akx(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_897
tff(fact_9078_ATP_Olambda__898,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_et(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_898
tff(fact_9079_ATP_Olambda__899,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_om(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,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub))) ) ).

% ATP.lambda_899
tff(fact_9080_ATP_Olambda__900,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_ft(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_900
tff(fact_9081_ATP_Olambda__901,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ana(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),aa(nat,real,Uua,Uub)) ) ) ).

% ATP.lambda_901
tff(fact_9082_ATP_Olambda__902,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tf(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_902
tff(fact_9083_ATP_Olambda__903,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,B),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_any(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),real_V7770717601297561774m_norm(B,aa(nat,B,Uua,Uub))) ) ) ).

% ATP.lambda_903
tff(fact_9084_ATP_Olambda__904,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ams(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( archim6421214686448440834_floor(B,aa(A,B,Uu,Uub)) = archim6421214686448440834_floor(B,Uua) ) ) ) ).

% ATP.lambda_904
tff(fact_9085_ATP_Olambda__905,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_amt(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( archimedean_ceiling(B,aa(A,B,Uu,Uub)) = archimedean_ceiling(B,Uua) ) ) ) ).

% ATP.lambda_905
tff(fact_9086_ATP_Olambda__906,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aoa(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua) ) ) ).

% ATP.lambda_906
tff(fact_9087_ATP_Olambda__907,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_ls(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_907
tff(fact_9088_ATP_Olambda__908,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_anh(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),ring_1_of_int(B,archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_908
tff(fact_9089_ATP_Olambda__909,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),$o)),Uua: list(A),Uub: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_ayx(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub)
        <=> ( ? [Y3: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = cons(A,Y3,Ys4) ) )
            | ? [X: A,Y3: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = cons(A,X,Xs3) )
                & ( Uub = cons(A,Y3,Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3) )
            | ? [X: A,Y3: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = cons(A,X,Xs3) )
                & ( Uub = cons(A,Y3,Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uu,Xs3),Ys4) ) ) ) ) ).

% ATP.lambda_909
tff(fact_9090_ATP_Olambda__910,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_oi(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),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_910
tff(fact_9091_ATP_Olambda__911,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_oj(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_911
tff(fact_9092_ATP_Olambda__912,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_oh(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_912
tff(fact_9093_ATP_Olambda__913,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_aub(A,fun(int,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),ring_1_of_int(A,Uua)),power_int(A,Uu,aa(int,int,aa(int,fun(int,int),minus_minus(int),Uua),one_one(int))))) ) ).

% ATP.lambda_913
tff(fact_9094_ATP_Olambda__914,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dk(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_914
tff(fact_9095_ATP_Olambda__915,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dj(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_915
tff(fact_9096_ATP_Olambda__916,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_ea(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),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_916
tff(fact_9097_ATP_Olambda__917,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_agq(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_917
tff(fact_9098_ATP_Olambda__918,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_cx(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_918
tff(fact_9099_ATP_Olambda__919,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dg(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_919
tff(fact_9100_ATP_Olambda__920,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_amx(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> aa(set(real),$o,member(real,Uub),set_or5935395276787703475ssThan(real,Uu,Uua)) ) ).

% ATP.lambda_920
tff(fact_9101_ATP_Olambda__921,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: real,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kr(real,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_921
tff(fact_9102_ATP_Olambda__922,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: real,Uub: nat] : aa(nat,A,aa(real,fun(nat,A),aTP_Lamp_ks(fun(nat,A),fun(real,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_922
tff(fact_9103_ATP_Olambda__923,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: real,Uub: A] : aa(A,B,aa(real,fun(A,B),aTP_Lamp_xz(fun(A,B),fun(real,fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_923
tff(fact_9104_ATP_Olambda__924,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_wh(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_924
tff(fact_9105_ATP_Olambda__925,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ur(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_925
tff(fact_9106_ATP_Olambda__926,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_uq(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_926
tff(fact_9107_ATP_Olambda__927,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ame(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_927
tff(fact_9108_ATP_Olambda__928,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_amn(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_928
tff(fact_9109_ATP_Olambda__929,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_amf(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_929
tff(fact_9110_ATP_Olambda__930,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_alr(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_930
tff(fact_9111_ATP_Olambda__931,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_vc(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_931
tff(fact_9112_ATP_Olambda__932,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_alp(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_932
tff(fact_9113_ATP_Olambda__933,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_alt(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_933
tff(fact_9114_ATP_Olambda__934,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_alo(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_934
tff(fact_9115_ATP_Olambda__935,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ack(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_935
tff(fact_9116_ATP_Olambda__936,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_eo(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_936
tff(fact_9117_ATP_Olambda__937,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_cb(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_937
tff(fact_9118_ATP_Olambda__938,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_gc(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_938
tff(fact_9119_ATP_Olambda__939,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_afn(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_939
tff(fact_9120_ATP_Olambda__940,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_ach(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_940
tff(fact_9121_ATP_Olambda__941,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ex(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_941
tff(fact_9122_ATP_Olambda__942,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_yd(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_942
tff(fact_9123_ATP_Olambda__943,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_apk(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_943
tff(fact_9124_ATP_Olambda__944,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_wg(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_944
tff(fact_9125_ATP_Olambda__945,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aaq(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_945
tff(fact_9126_ATP_Olambda__946,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_acw(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_946
tff(fact_9127_ATP_Olambda__947,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_adw(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_947
tff(fact_9128_ATP_Olambda__948,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aik(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_948
tff(fact_9129_ATP_Olambda__949,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_sq(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_949
tff(fact_9130_ATP_Olambda__950,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_hx(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_950
tff(fact_9131_ATP_Olambda__951,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_agy(fun(A,nat),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_951
tff(fact_9132_ATP_Olambda__952,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_awg(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_952
tff(fact_9133_ATP_Olambda__953,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_awn(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_953
tff(fact_9134_ATP_Olambda__954,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_th(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_954
tff(fact_9135_ATP_Olambda__955,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ug(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_955
tff(fact_9136_ATP_Olambda__956,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_tq(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_956
tff(fact_9137_ATP_Olambda__957,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adz(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_957
tff(fact_9138_ATP_Olambda__958,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uo(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_958
tff(fact_9139_ATP_Olambda__959,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: filter(B),Uua: fun(A,filter(C)),Uub: A] : aa(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_arm(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,Uu,aa(A,filter(C),Uua,Uub)) ).

% ATP.lambda_959
tff(fact_9140_ATP_Olambda__960,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,filter(C)),Uub: B] : aa(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_ara(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),Uu),Uua),Uub) = filtercomap(A,C,Uu,aa(B,filter(C),Uua,Uub)) ).

% ATP.lambda_960
tff(fact_9141_ATP_Olambda__961,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aqz(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtercomap(A,B,Uu,aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_961
tff(fact_9142_ATP_Olambda__962,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: set(product_prod(A,C)),Uua: fun(D,set(product_prod(C,B))),Uub: D] : aa(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_axq(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,Uu,aa(D,set(product_prod(C,B)),Uua,Uub)) ).

% ATP.lambda_962
tff(fact_9143_ATP_Olambda__963,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ato(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtermap(B,A,Uu,aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_963
tff(fact_9144_ATP_Olambda__964,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_aoz(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_964
tff(fact_9145_ATP_Olambda__965,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_abx(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_965
tff(fact_9146_ATP_Olambda__966,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_adk(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ).

% ATP.lambda_966
tff(fact_9147_ATP_Olambda__967,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_alj($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_967
tff(fact_9148_ATP_Olambda__968,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_rj(fun(B,set(A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu,Uua)) ) ).

% ATP.lambda_968
tff(fact_9149_ATP_Olambda__969,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_rq(B,fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert(B,Uu),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_969
tff(fact_9150_ATP_Olambda__970,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_rp(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),insert(A,Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_970
tff(fact_9151_ATP_Olambda__971,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_sj(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_971
tff(fact_9152_ATP_Olambda__972,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(A)),Uub: C] : aa(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_ul(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),image(A,B,Uu),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_972
tff(fact_9153_ATP_Olambda__973,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_aom(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),Uu),Uua),Uub) = aa(set(B),set(C),image(B,C,Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_973
tff(fact_9154_ATP_Olambda__974,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ali($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_974
tff(fact_9155_ATP_Olambda__975,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_azm(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_975
tff(fact_9156_ATP_Olambda__976,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_atk(fun(A,B),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_976
tff(fact_9157_ATP_Olambda__977,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ig(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,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_977
tff(fact_9158_ATP_Olambda__978,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_anf(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_978
tff(fact_9159_ATP_Olambda__979,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_auu(set(B),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uu),Uua) ).

% ATP.lambda_979
tff(fact_9160_ATP_Olambda__980,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_axa(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert(B,Uu),Uua) ).

% ATP.lambda_980
tff(fact_9161_ATP_Olambda__981,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_avg(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image(D,B,Uu),Uua) ).

% ATP.lambda_981
tff(fact_9162_ATP_Olambda__982,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_ib(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_982
tff(fact_9163_ATP_Olambda__983,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_ds(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_983
tff(fact_9164_ATP_Olambda__984,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_agc(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_984
tff(fact_9165_ATP_Olambda__985,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aft(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uua)) ) ).

% ATP.lambda_985
tff(fact_9166_ATP_Olambda__986,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_aeu(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_986
tff(fact_9167_ATP_Olambda__987,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_wo(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_987
tff(fact_9168_ATP_Olambda__988,axiom,
    ! [Uu: fun(real,$o),Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_amu(fun(real,$o),fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,Uu,aa(real,real,aa(real,fun(real,real),plus_plus(real),Uub),Uua)) ) ).

% ATP.lambda_988
tff(fact_9169_ATP_Olambda__989,axiom,
    ! [A: $tType,Uu: fun(real,A),Uua: real,Uub: real] : aa(real,A,aa(real,fun(real,A),aTP_Lamp_aii(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_989
tff(fact_9170_ATP_Olambda__990,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aki(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_990
tff(fact_9171_ATP_Olambda__991,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_st(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu),Uua),Uub) = aa(nat,set(A),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_991
tff(fact_9172_ATP_Olambda__992,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_en(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_992
tff(fact_9173_ATP_Olambda__993,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afs(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_993
tff(fact_9174_ATP_Olambda__994,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_anp(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_994
tff(fact_9175_ATP_Olambda__995,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_hy(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_995
tff(fact_9176_ATP_Olambda__996,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_cw(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_996
tff(fact_9177_ATP_Olambda__997,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_amc(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_997
tff(fact_9178_ATP_Olambda__998,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_aez(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_998
tff(fact_9179_ATP_Olambda__999,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_afi(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_999
tff(fact_9180_ATP_Olambda__1000,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_vr(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_1000
tff(fact_9181_ATP_Olambda__1001,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_na(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(A,B),$o,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).

% ATP.lambda_1001
tff(fact_9182_ATP_Olambda__1002,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_bf(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_1002
tff(fact_9183_ATP_Olambda__1003,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_are(fun(product_prod(A,A),$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(product_prod(A,A),$o,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)) ) ) ).

% ATP.lambda_1003
tff(fact_9184_ATP_Olambda__1004,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_afb(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),Uub)) ) ).

% ATP.lambda_1004
tff(fact_9185_ATP_Olambda__1005,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_acm(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_1005
tff(fact_9186_ATP_Olambda__1006,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_ge(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_1006
tff(fact_9187_ATP_Olambda__1007,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_yw(fun(real,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,Uu,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_1007
tff(fact_9188_ATP_Olambda__1008,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(real,A),Uua: fun(nat,real),Uub: nat] : aa(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_ajg(fun(real,A),fun(fun(nat,real),fun(nat,A)),Uu),Uua),Uub) = aa(real,A,Uu,aa(nat,real,Uua,Uub)) ) ).

% ATP.lambda_1008
tff(fact_9189_ATP_Olambda__1009,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,nat),Uub: nat] : aa(nat,real,aa(fun(nat,nat),fun(nat,real),aTP_Lamp_asr(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),Uu),Uua),Uub) = aa(nat,real,Uu,aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_1009
tff(fact_9190_ATP_Olambda__1010,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_anq(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_1010
tff(fact_9191_ATP_Olambda__1011,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_ast(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_1011
tff(fact_9192_ATP_Olambda__1012,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: A] : aa(A,fun(B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_qf(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),Uu),Uua),Uub) = aa(D,fun(B,C),Uu,aa(A,D,Uua,Uub)) ).

% ATP.lambda_1012
tff(fact_9193_ATP_Olambda__1013,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,set(A)),Uua: fun(B,C),Uub: B] : aa(B,set(A),aa(fun(B,C),fun(B,set(A)),aTP_Lamp_ro(fun(C,set(A)),fun(fun(B,C),fun(B,set(A))),Uu),Uua),Uub) = aa(C,set(A),Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_1013
tff(fact_9194_ATP_Olambda__1014,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_tl(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_1014
tff(fact_9195_ATP_Olambda__1015,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(num,B),Uub: num] : aa(num,A,aa(fun(num,B),fun(num,A),aTP_Lamp_bal(fun(B,A),fun(fun(num,B),fun(num,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(num,B,Uua,Uub)) ).

% ATP.lambda_1015
tff(fact_9196_ATP_Olambda__1016,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_pe(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_1016
tff(fact_9197_ATP_Olambda__1017,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_mr(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_1017
tff(fact_9198_ATP_Olambda__1018,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_asm(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1018
tff(fact_9199_ATP_Olambda__1019,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aod(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_1019
tff(fact_9200_ATP_Olambda__1020,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(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_aoi(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_1020
tff(fact_9201_ATP_Olambda__1021,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_ako(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_1021
tff(fact_9202_ATP_Olambda__1022,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_afz(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_1022
tff(fact_9203_ATP_Olambda__1023,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_ahn(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_1023
tff(fact_9204_ATP_Olambda__1024,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_yl(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1024
tff(fact_9205_ATP_Olambda__1025,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_aoq(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1025
tff(fact_9206_ATP_Olambda__1026,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_ahl(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1026
tff(fact_9207_ATP_Olambda__1027,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_ahk(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1027
tff(fact_9208_ATP_Olambda__1028,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_aga(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_1028
tff(fact_9209_ATP_Olambda__1029,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_aqd(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1029
tff(fact_9210_ATP_Olambda__1030,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_aex(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1030
tff(fact_9211_ATP_Olambda__1031,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_vs(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_1031
tff(fact_9212_ATP_Olambda__1032,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ash(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1032
tff(fact_9213_ATP_Olambda__1033,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ats(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_1033
tff(fact_9214_ATP_Olambda__1034,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_acl(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1034
tff(fact_9215_ATP_Olambda__1035,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( order(C)
        & order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_asu(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_1035
tff(fact_9216_ATP_Olambda__1036,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_akr(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_1036
tff(fact_9217_ATP_Olambda__1037,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_acn(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_1037
tff(fact_9218_ATP_Olambda__1038,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,$o),Uub: nat] :
      ( aa(nat,$o,aa(fun(nat,$o),fun(nat,$o),aTP_Lamp_ata(fun(nat,nat),fun(fun(nat,$o),fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_1038
tff(fact_9219_ATP_Olambda__1039,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,nat),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_atd(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_1039
tff(fact_9220_ATP_Olambda__1040,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,nat),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ate(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_1040
tff(fact_9221_ATP_Olambda__1041,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_zz(fun(C,A),fun(fun(A,B),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(C,A,Uu,Uub)) ) ).

% ATP.lambda_1041
tff(fact_9222_ATP_Olambda__1042,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_ud(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_1042
tff(fact_9223_ATP_Olambda__1043,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_xt(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1043
tff(fact_9224_ATP_Olambda__1044,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_afe(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1044
tff(fact_9225_ATP_Olambda__1045,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_ami(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1045
tff(fact_9226_ATP_Olambda__1046,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_vt(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_1046
tff(fact_9227_ATP_Olambda__1047,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B)
        & topolo7287701948861334536_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_atn(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1047
tff(fact_9228_ATP_Olambda__1048,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_ail(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1048
tff(fact_9229_ATP_Olambda__1049,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_zy(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1049
tff(fact_9230_ATP_Olambda__1050,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_aku(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1050
tff(fact_9231_ATP_Olambda__1051,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_im(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1051
tff(fact_9232_ATP_Olambda__1052,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_in(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1052
tff(fact_9233_ATP_Olambda__1053,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_pr(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_1053
tff(fact_9234_ATP_Olambda__1054,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aqy(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ).

% ATP.lambda_1054
tff(fact_9235_ATP_Olambda__1055,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: D,Uub: A] : aa(A,set(B),aa(D,fun(A,set(B)),aTP_Lamp_avh(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).

% ATP.lambda_1055
tff(fact_9236_ATP_Olambda__1056,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_arq(fun(B,$o),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_1056
tff(fact_9237_ATP_Olambda__1057,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_asz(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_1057
tff(fact_9238_ATP_Olambda__1058,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),$o),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),$o),aa(A,fun(A,fun(product_prod(A,A),$o)),aTP_Lamp_arw(fun(product_prod(A,A),$o),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_arv(fun(product_prod(A,A),$o),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ) ).

% ATP.lambda_1058
tff(fact_9239_ATP_Olambda__1059,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_acd(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_acc(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_1059
tff(fact_9240_ATP_Olambda__1060,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_abp(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_abo(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub)) ) ).

% ATP.lambda_1060
tff(fact_9241_ATP_Olambda__1061,axiom,
    ! [B: $tType,C: $tType,A: $tType,E3: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E3,set(A))))),Uua: set(product_prod(D,E3)),Uub: product_prod(B,C)] : aa(product_prod(B,C),set(A),aa(set(product_prod(D,E3)),fun(product_prod(B,C),set(A)),aTP_Lamp_sz(fun(B,fun(C,fun(D,fun(E3,set(A))))),fun(set(product_prod(D,E3)),fun(product_prod(B,C),set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D,E3)),set(set(A)),image(product_prod(D,E3),set(A),aa(product_prod(B,C),fun(product_prod(D,E3),set(A)),aTP_Lamp_sy(fun(B,fun(C,fun(D,fun(E3,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E3),set(A))),Uu),Uub)),Uua)) ).

% ATP.lambda_1061
tff(fact_9242_ATP_Olambda__1062,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_sf(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aa(C,fun(B,A),aTP_Lamp_rm(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_1062
tff(fact_9243_ATP_Olambda__1063,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_rn(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aa(C,fun(B,A),aTP_Lamp_rm(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_1063
tff(fact_9244_ATP_Olambda__1064,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: set(A),Uub: A] : aa(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aiq(A,fun(set(A),fun(A,filter(A))),Uu),Uua),Uub) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),Uub)),Uua)),aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_1064
tff(fact_9245_ATP_Olambda__1065,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: set(A),Uub: A] : aa(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aip(A,fun(set(A),fun(A,filter(A))),Uu),Uua),Uub) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),Uub)),Uua)),aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_1065
tff(fact_9246_ATP_Olambda__1066,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: set(A),Uub: set(A)] : aa(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_aje(A,fun(set(A),fun(set(A),filter(A))),Uu),Uua),Uub) = aa(set(A),filter(A),principal(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uub),Uua)),aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_1066
tff(fact_9247_ATP_Olambda__1067,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_fi(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_1067
tff(fact_9248_ATP_Olambda__1068,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_ie(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_1068
tff(fact_9249_ATP_Olambda__1069,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aor(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_1069
tff(fact_9250_ATP_Olambda__1070,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_se(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_1070
tff(fact_9251_ATP_Olambda__1071,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_rl(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_1071
tff(fact_9252_ATP_Olambda__1072,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ahd(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_1072
tff(fact_9253_ATP_Olambda__1073,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_amh(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_1073
tff(fact_9254_ATP_Olambda__1074,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akp(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_1074
tff(fact_9255_ATP_Olambda__1075,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_amj(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_1075
tff(fact_9256_ATP_Olambda__1076,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t1_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akw(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_1076
tff(fact_9257_ATP_Olambda__1077,axiom,
    ! [C: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: fun(C,A),Uub: C] :
          ( aa(C,$o,aa(fun(C,A),fun(C,$o),aTP_Lamp_amg(A,fun(fun(C,A),fun(C,$o)),Uu),Uua),Uub)
        <=> ( aa(C,A,Uua,Uub) != Uu ) ) ) ).

% ATP.lambda_1077
tff(fact_9258_ATP_Olambda__1078,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,set(A)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_sd(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),Uu),aa(B,set(C),Uua,Uub))) ).

% ATP.lambda_1078
tff(fact_9259_ATP_Olambda__1079,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_si(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),Uu),aa(C,set(B),Uua,Uub))) ).

% ATP.lambda_1079
tff(fact_9260_ATP_Olambda__1080,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_sg(fun(B,A),fun(fun(C,set(B)),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,Uu),aa(C,set(B),Uua,Uub))) ) ).

% ATP.lambda_1080
tff(fact_9261_ATP_Olambda__1081,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aon(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_1081
tff(fact_9262_ATP_Olambda__1082,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,set(A)),Uua: fun(B,set(C)),Uub: B] : aa(B,set(A),aa(fun(B,set(C)),fun(B,set(A)),aTP_Lamp_sm(fun(C,set(A)),fun(fun(B,set(C)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),Uu),aa(B,set(C),Uua,Uub))) ).

% ATP.lambda_1082
tff(fact_9263_ATP_Olambda__1083,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aqr(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_1083
tff(fact_9264_ATP_Olambda__1084,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_up(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_1084
tff(fact_9265_ATP_Olambda__1085,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_avj(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image(D,set(B),Uu),Uua)) ).

% ATP.lambda_1085
tff(fact_9266_ATP_Olambda__1086,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_pi(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_cj(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_1086
tff(fact_9267_ATP_Olambda__1087,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(B,real),Uua: fun(real,A),Uub: B] : aa(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_zs(fun(B,real),fun(fun(real,A),fun(B,real)),Uu),Uua),Uub) = ring_1_of_int(real,archim6421214686448440834_floor(A,aa(real,A,Uua,aa(B,real,Uu,Uub)))) ) ).

% ATP.lambda_1087
tff(fact_9268_ATP_Olambda__1088,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,B),fun(product_prod(A,B),$o),aTP_Lamp_azl(set(A),fun(fun(A,B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [A5: A] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),aa(A,B,Uua,A5)) )
          & aa(set(A),$o,member(A,A5),Uu) ) ) ).

% ATP.lambda_1088
tff(fact_9269_ATP_Olambda__1089,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bay(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A5: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A5) )
              & aa(set(A),$o,member(A,A5),Uu) ) ) ) ).

% ATP.lambda_1089
tff(fact_9270_ATP_Olambda__1090,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: A,Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bam(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [B5: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),B5) )
              & aa(set(A),$o,member(A,B5),Uua) ) ) ) ).

% ATP.lambda_1090
tff(fact_9271_ATP_Olambda__1091,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bat(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A5: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A5) )
              & aa(set(A),$o,member(A,A5),Uu) ) ) ) ).

% ATP.lambda_1091
tff(fact_9272_ATP_Olambda__1092,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ayq(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( ( Uub = aa(B,A,Uu,X) )
          & aa(set(B),$o,member(B,X),Uua) ) ) ).

% ATP.lambda_1092
tff(fact_9273_ATP_Olambda__1093,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,set(C)),Uua: fun(C,$o),Uub: B] :
      ( aa(B,$o,aa(fun(C,$o),fun(B,$o),aTP_Lamp_ayv(fun(B,set(C)),fun(fun(C,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X: C] :
          ( aa(set(C),$o,member(C,X),aa(B,set(C),Uu,Uub))
         => aa(C,$o,Uua,X) ) ) ).

% ATP.lambda_1093
tff(fact_9274_ATP_Olambda__1094,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,$o),Uub: set(A)] :
      ( aa(set(A),$o,aa(fun(B,$o),fun(set(A),$o),aTP_Lamp_azc(fun(B,set(A)),fun(fun(B,$o),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( ( Uub = aa(B,set(A),Uu,X) )
          & aa(B,$o,Uua,X) ) ) ).

% ATP.lambda_1094
tff(fact_9275_ATP_Olambda__1095,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_ayp(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( ( Uub = aa(B,A,Uu,X) )
          & aa(B,$o,Uua,X) ) ) ).

% ATP.lambda_1095
tff(fact_9276_ATP_Olambda__1096,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,B),Uub: B] :
      ( aa(B,$o,aa(fun(A,B),fun(B,$o),aTP_Lamp_azh(fun(A,$o),fun(fun(A,B),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X: A] :
          ( ( Uub = aa(A,B,Uua,X) )
          & aa(A,$o,Uu,X) ) ) ).

% ATP.lambda_1096
tff(fact_9277_ATP_Olambda__1097,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] :
      ( aa(A,$o,aa(fun(A,fun(B,$o)),fun(A,$o),aTP_Lamp_ayr(set(B),fun(fun(A,fun(B,$o)),fun(A,$o)),Uu),Uua),Uub)
    <=> ! [X: B] :
          ( aa(set(B),$o,member(B,X),Uu)
         => aa(B,$o,aa(A,fun(B,$o),Uua,Uub),X) ) ) ).

% ATP.lambda_1097
tff(fact_9278_ATP_Olambda__1098,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_azj(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X: A] :
          ( aa(set(A),$o,member(A,X),Uu)
         => aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X) ) ) ).

% ATP.lambda_1098
tff(fact_9279_ATP_Olambda__1099,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_ank(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),N4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,Uub,N4)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_1099
tff(fact_9280_ATP_Olambda__1100,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ays(fun(B,set(A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ! [X: B] :
          ( aa(set(B),$o,member(B,X),Uua)
         => aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu,X)) ) ) ).

% ATP.lambda_1100
tff(fact_9281_ATP_Olambda__1101,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aob(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [A5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),A5)
             => ! [B5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A5),B5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or3652927894154168847AtMost(nat,A5,B5)))),aa(nat,real,Uua,A5)) ) ) ) ) ).

% ATP.lambda_1101
tff(fact_9282_ATP_Olambda__1102,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_ayj(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [Y3: A] :
          ( aa(A,$o,Uu,Y3)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),Y3) ) ) ).

% ATP.lambda_1102
tff(fact_9283_ATP_Olambda__1103,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_azd(fun(B,set(A)),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( aa(B,$o,Uua,X)
          & aa(set(A),$o,member(A,Uub),aa(B,set(A),Uu,X)) ) ) ).

% ATP.lambda_1103
tff(fact_9284_ATP_Olambda__1104,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_azu(fun(A,A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ? [N4: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N4),Uu),Uua) ) ).

% ATP.lambda_1104
tff(fact_9285_ATP_Olambda__1105,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_bbb(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [A5: A,V5: list(A)] :
          ( ( Uub = append(A,Uua,cons(A,A5,V5)) )
          | ? [U4: list(A),Aa3: A,B5: A,Va4: list(A),W3: list(A)] :
              ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa3),B5)),Uu)
              & ( Uua = append(A,U4,cons(A,Aa3,Va4)) )
              & ( Uub = append(A,U4,cons(A,B5,W3)) ) ) ) ) ).

% ATP.lambda_1105
tff(fact_9286_ATP_Olambda__1106,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bax(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A5: A,B5: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B5) )
              & aa(set(A),$o,member(A,A5),Uu)
              & aa(set(A),$o,member(A,B5),Uua) ) ) ) ).

% ATP.lambda_1106
tff(fact_9287_ATP_Olambda__1107,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bau(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A5: A,B5: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B5) )
              & aa(set(A),$o,member(A,A5),Uu)
              & aa(set(A),$o,member(A,B5),Uua) ) ) ) ).

% ATP.lambda_1107
tff(fact_9288_ATP_Olambda__1108,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(list(A)),fun(list(A),$o),aTP_Lamp_ayw(set(A),fun(set(list(A)),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [X: A,Xs3: list(A)] :
          ( ( Uub = cons(A,X,Xs3) )
          & aa(set(A),$o,member(A,X),Uu)
          & aa(set(list(A)),$o,member(list(A),Xs3),Uua) ) ) ).

% ATP.lambda_1108
tff(fact_9289_ATP_Olambda__1109,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_bbc(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [Us3: list(A),Z3: A,Z7: A,Vs3: list(A)] :
          ( ( Uua = append(A,Us3,cons(A,Z3,Vs3)) )
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Z7)),Uu)
          & ( Uub = append(A,Us3,cons(A,Z7,Vs3)) ) ) ) ).

% ATP.lambda_1109
tff(fact_9290_ATP_Olambda__1110,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_mj(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_1110
tff(fact_9291_ATP_Olambda__1111,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_ko(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub)
            & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),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),aa(num,num,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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_1111
tff(fact_9292_ATP_Olambda__1112,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_kk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),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),aa(num,num,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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_1112
tff(fact_9293_ATP_Olambda__1113,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_km(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,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),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),aa(num,num,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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_1113
tff(fact_9294_ATP_Olambda__1114,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] :
      aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_axn(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = $ite(aa(product_prod(C,A),A,product_snd(C,A),Uu) = Uua,aa(set(product_prod(C,B)),set(product_prod(C,B)),insert(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_1114
tff(fact_9295_ATP_Olambda__1115,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] :
      aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_awz(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(set(B),$o,member(B,Uuc),aa(set(A),set(B),image(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_1115
tff(fact_9296_ATP_Olambda__1116,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: A,Uub: B,Uuc: set(B)] :
      aa(set(B),set(B),aa(B,fun(set(B),set(B)),aa(A,fun(B,fun(set(B),set(B))),aTP_Lamp_aya(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = $ite(aa(set(A),$o,member(A,Uua),Uu),aa(set(B),set(B),insert(B,Uub),Uuc),Uuc) ).

% ATP.lambda_1116
tff(fact_9297_ATP_Olambda__1117,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_jy(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_1117
tff(fact_9298_ATP_Olambda__1118,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_ka(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_1118
tff(fact_9299_ATP_Olambda__1119,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_aim(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1119
tff(fact_9300_ATP_Olambda__1120,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_kb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_1120
tff(fact_9301_ATP_Olambda__1121,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_jz(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_1121
tff(fact_9302_ATP_Olambda__1122,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: fun(A,B),Uuc: A] :
      aa(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_awu(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(set(A),$o,member(A,Uuc),Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_1122
tff(fact_9303_ATP_Olambda__1123,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_gh(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(set(nat),$o,member(nat,Uuc),Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_1123
tff(fact_9304_ATP_Olambda__1124,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(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_id(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1124
tff(fact_9305_ATP_Olambda__1125,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(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_fr(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1125
tff(fact_9306_ATP_Olambda__1126,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_pt(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_1126
tff(fact_9307_ATP_Olambda__1127,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),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_awd(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_1127
tff(fact_9308_ATP_Olambda__1128,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,$o),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_uk(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1128
tff(fact_9309_ATP_Olambda__1129,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,$o),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_uj(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1129
tff(fact_9310_ATP_Olambda__1130,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_aes(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_1130
tff(fact_9311_ATP_Olambda__1131,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
      aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_ais(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_1131
tff(fact_9312_ATP_Olambda__1132,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: fun(A,$o),Uuc: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aa(fun(A,B),fun(fun(A,$o),fun(A,B)),aTP_Lamp_awk(fun(A,B),fun(fun(A,B),fun(fun(A,$o),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uub,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uua,Uuc)) ) ).

% ATP.lambda_1132
tff(fact_9313_ATP_Olambda__1133,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_axw(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_axv(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_1133
tff(fact_9314_ATP_Olambda__1134,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_axu(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_axt(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_1134
tff(fact_9315_ATP_Olambda__1135,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(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_qk(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uuc),aa(A,fun(A,A),times_times(A),Uua)),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))) ) ).

% ATP.lambda_1135
tff(fact_9316_ATP_Olambda__1136,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: fun(D,B),Uuc: D] : aa(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_avd(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),aa(D,B,Uub,Uuc)) ) ).

% ATP.lambda_1136
tff(fact_9317_ATP_Olambda__1137,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_bad(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_1137
tff(fact_9318_ATP_Olambda__1138,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_dp(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_do(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_1138
tff(fact_9319_ATP_Olambda__1139,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_vo($o,fun($o,fun(code_integer,fun($o,char))),(Uu)),(Uua)),Uub),(Uuc)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_vn($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu)),(Uua)),(Uuc))),code_bit_cut_integer(Uub)) ).

% ATP.lambda_1139
tff(fact_9320_ATP_Olambda__1140,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_ce(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_cd(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_1140
tff(fact_9321_ATP_Olambda__1141,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_kc(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(
                Uuc = zero_zero(nat),
                aa(A,A,uminus_uminus(A),Uub),
                $ite(Uuc = Uu,one_one(A),zero_zero(A)) )),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_1141
tff(fact_9322_ATP_Olambda__1142,axiom,
    ! [C: $tType,A: $tType,B: $tType,E3: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E3,set(A))))),Uua: product_prod(D,E3),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(product_prod(D,E3),fun(B,fun(C,set(A))),aTP_Lamp_sx(fun(B,fun(C,fun(D,fun(E3,set(A))))),fun(product_prod(D,E3),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(product_prod(D,E3),set(A),aa(fun(D,fun(E3,set(A))),fun(product_prod(D,E3),set(A)),product_case_prod(D,E3,set(A)),aa(C,fun(D,fun(E3,set(A))),aa(B,fun(C,fun(D,fun(E3,set(A)))),Uu,Uub),Uuc)),Uua) ).

% ATP.lambda_1142
tff(fact_9323_ATP_Olambda__1143,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_hs(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_hr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_av(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1143
tff(fact_9324_ATP_Olambda__1144,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_cm(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_cl(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_av(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1144
tff(fact_9325_ATP_Olambda__1145,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_jw(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_jv(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_1145
tff(fact_9326_ATP_Olambda__1146,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_jr(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_jq(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_1146
tff(fact_9327_ATP_Olambda__1147,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_xm(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_1147
tff(fact_9328_ATP_Olambda__1148,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_xk(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_1148
tff(fact_9329_ATP_Olambda__1149,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_xi(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,aa(real,fun(real,real),minus_minus(real),Uub),Uua)),Uuc)) ).

% ATP.lambda_1149
tff(fact_9330_ATP_Olambda__1150,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_xj(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,aa(real,fun(real,real),minus_minus(real),Uua),Uub)),Uuc)) ).

% ATP.lambda_1150
tff(fact_9331_ATP_Olambda__1151,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_dw(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(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,aa(nat,fun(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_1151
tff(fact_9332_ATP_Olambda__1152,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qn(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = 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),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).

% ATP.lambda_1152
tff(fact_9333_ATP_Olambda__1153,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_do(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,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Uuc))) ) ).

% ATP.lambda_1153
tff(fact_9334_ATP_Olambda__1154,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_aha(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,aa(A,fun(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_1154
tff(fact_9335_ATP_Olambda__1155,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_bao(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
          & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uua) ) ) ) ).

% ATP.lambda_1155
tff(fact_9336_ATP_Olambda__1156,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: fun(A,B),Uua: set(B),Uub: B,Uuc: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_akz(fun(A,B),fun(set(B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(B),$o,member(B,aa(A,B,Uu,Uuc)),Uua)
            & ( aa(A,B,Uu,Uuc) != Uub ) ) ) ) ).

% ATP.lambda_1156
tff(fact_9337_ATP_Olambda__1157,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_alu(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(B),$o,member(B,aa(A,B,Uu,Uuc)),Uub)
            & ( aa(A,B,Uu,Uuc) != Uua ) ) ) ) ).

% ATP.lambda_1157
tff(fact_9338_ATP_Olambda__1158,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_anl(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_1158
tff(fact_9339_ATP_Olambda__1159,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_amz(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_1159
tff(fact_9340_ATP_Olambda__1160,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_jt(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_1160
tff(fact_9341_ATP_Olambda__1161,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,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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_1161
tff(fact_9342_ATP_Olambda__1162,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_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))),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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_1162
tff(fact_9343_ATP_Olambda__1163,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_lq(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_1163
tff(fact_9344_ATP_Olambda__1164,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_ban(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys: list(A),X: A,Y3: A,Xs6: list(A),Ys7: list(A)] :
            ( ( Uub = append(A,Xys,cons(A,X,Xs6)) )
            & ( Uuc = append(A,Xys,cons(A,Y3,Ys7)) )
            & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),Uu) ) ) ) ).

% ATP.lambda_1164
tff(fact_9345_ATP_Olambda__1165,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_dz(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,aa(nat,fun(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_1165
tff(fact_9346_ATP_Olambda__1166,axiom,
    ! [A: $tType,Uu: $o,Uua: A,Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_mc($o,fun(A,fun(A,fun(A,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_1166
tff(fact_9347_ATP_Olambda__1167,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_ms(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),aa(set(B),set(A),image(B,A,Uu),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_1167
tff(fact_9348_ATP_Olambda__1168,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_cj(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1168
tff(fact_9349_ATP_Olambda__1169,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_av(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_1169
tff(fact_9350_ATP_Olambda__1170,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_dx(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,aa(A,fun(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_1170
tff(fact_9351_ATP_Olambda__1171,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: set(A),Uua: fun(A,A),Uub: fun(A,A),Uuc: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_alv(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu)
           => ( aa(A,A,Uua,Uuc) = aa(A,A,Uub,Uuc) ) ) ) ) ).

% ATP.lambda_1171
tff(fact_9352_ATP_Olambda__1172,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: fun(A,real),Uuc: A] :
          ( aa(A,$o,aa(fun(A,real),fun(A,$o),aa(fun(A,real),fun(fun(A,real),fun(A,$o)),aTP_Lamp_aqg(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uua,Uuc)),aa(A,real,Uub,Uuc)) ) ) ) ).

% ATP.lambda_1172
tff(fact_9353_ATP_Olambda__1173,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_mu(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_1173
tff(fact_9354_ATP_Olambda__1174,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(B,A),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_vb(set(B),fun(fun(B,A),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uu)
        & ( aa(B,A,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_1174
tff(fact_9355_ATP_Olambda__1175,axiom,
    ! [A: $tType,C: $tType,Uu: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_mx(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_1175
tff(fact_9356_ATP_Olambda__1176,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_nb(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,member(A,Uuc),Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_1176
tff(fact_9357_ATP_Olambda__1177,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_asw(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uub)
        & aa(set(A),$o,member(A,aa(B,A,Uua,Uuc)),Uu) ) ) ).

% ATP.lambda_1177
tff(fact_9358_ATP_Olambda__1178,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_dy(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,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uuc))) ) ).

% ATP.lambda_1178
tff(fact_9359_ATP_Olambda__1179,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_ji(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_1179
tff(fact_9360_ATP_Olambda__1180,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_nl(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),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_1180
tff(fact_9361_ATP_Olambda__1181,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_qt(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_1181
tff(fact_9362_ATP_Olambda__1182,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_ni(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),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_1182
tff(fact_9363_ATP_Olambda__1183,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_qr(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_1183
tff(fact_9364_ATP_Olambda__1184,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qv(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = 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),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ).

% ATP.lambda_1184
tff(fact_9365_ATP_Olambda__1185,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qx(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = 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),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).

% ATP.lambda_1185
tff(fact_9366_ATP_Olambda__1186,axiom,
    ! [Uu: set(complex),Uua: complex,Uub: complex,Uuc: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aa(complex,fun(complex,fun(complex,$o)),aTP_Lamp_arn(set(complex),fun(complex,fun(complex,fun(complex,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uub = Uua )
       => aa(set(complex),$o,member(complex,Uuc),Uu) ) ) ).

% ATP.lambda_1186
tff(fact_9367_ATP_Olambda__1187,axiom,
    ! [Uu: set(real),Uua: real,Uub: real,Uuc: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aa(real,fun(real,fun(real,$o)),aTP_Lamp_aro(set(real),fun(real,fun(real,fun(real,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uub = Uua )
       => aa(set(real),$o,member(real,Uuc),Uu) ) ) ).

% ATP.lambda_1187
tff(fact_9368_ATP_Olambda__1188,axiom,
    ! [A: $tType] :
      ( topolo569519726778239578ormity(A)
     => ! [Uu: set(A),Uua: A,Uub: A,Uuc: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_arf(set(A),fun(A,fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( ( Uub = Uua )
           => aa(set(A),$o,member(A,Uuc),Uu) ) ) ) ).

% ATP.lambda_1188
tff(fact_9369_ATP_Olambda__1189,axiom,
    ! [A: $tType,B: $tType,Uu: filter(A),Uua: filter(B),Uub: fun(A,$o),Uuc: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(fun(A,$o),fun(fun(B,$o),$o),aa(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o)),aTP_Lamp_arz(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> ( eventually(A,Uub,Uu)
        & eventually(B,Uuc,Uua) ) ) ).

% ATP.lambda_1189
tff(fact_9370_ATP_Olambda__1190,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_mb(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_1190
tff(fact_9371_ATP_Olambda__1191,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_mt(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,member(B,Uuc),Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_1191
tff(fact_9372_ATP_Olambda__1192,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A,Uuc: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_arg(fun(A,$o),fun(A,fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( ( Uub = Uua )
           => aa(A,$o,Uu,Uuc) ) ) ) ).

% ATP.lambda_1192
tff(fact_9373_ATP_Olambda__1193,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_rh(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(set(product_prod(A,B)),$o,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)),aa(C,set(product_prod(A,B)),Uu,Uua)) ) ).

% ATP.lambda_1193
tff(fact_9374_ATP_Olambda__1194,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ay(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_1194
tff(fact_9375_ATP_Olambda__1195,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ba(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu)
            & ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_1195
tff(fact_9376_ATP_Olambda__1196,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_ahi(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,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(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,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))) ) ).

% ATP.lambda_1196
tff(fact_9377_ATP_Olambda__1197,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_fp(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_1197
tff(fact_9378_ATP_Olambda__1198,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_op(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_1198
tff(fact_9379_ATP_Olambda__1199,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,$o),Uub: fun(A,B),Uuc: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_aok(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(B,$o,Uua,aa(A,B,Uub,Uuc))
        & aa(set(A),$o,member(A,Uuc),Uu) ) ) ).

% ATP.lambda_1199
tff(fact_9380_ATP_Olambda__1200,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_auz(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1200
tff(fact_9381_ATP_Olambda__1201,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,Uu: fun(C,set(A)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(product_prod(A,B)),aa(C,fun(D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_avi(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = product_Sigma(A,B,aa(C,set(A),Uu,Uub),aa(D,fun(A,set(B)),aTP_Lamp_avh(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).

% ATP.lambda_1201
tff(fact_9382_ATP_Olambda__1202,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_acc(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,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(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,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_1202
tff(fact_9383_ATP_Olambda__1203,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A,Uuc: nat] : aa(nat,B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_abo(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uua,Uuc)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_1203
tff(fact_9384_ATP_Olambda__1204,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_zn(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),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_1204
tff(fact_9385_ATP_Olambda__1205,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_ki(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,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_1205
tff(fact_9386_ATP_Olambda__1206,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_ane(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(set(B),$o,member(B,aa(A,B,Uu,Uuc)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Uub),aa(set(B),set(B),insert(B,Uua),bot_bot(set(B))))) ) ) ).

% ATP.lambda_1206
tff(fact_9387_ATP_Olambda__1207,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_jv(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_1207
tff(fact_9388_ATP_Olambda__1208,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_jn(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_1208
tff(fact_9389_ATP_Olambda__1209,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_jq(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,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_1209
tff(fact_9390_ATP_Olambda__1210,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_cd(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_1210
tff(fact_9391_ATP_Olambda__1211,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_awr(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_1211
tff(fact_9392_ATP_Olambda__1212,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_awo(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_1212
tff(fact_9393_ATP_Olambda__1213,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_ua(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_1213
tff(fact_9394_ATP_Olambda__1214,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_to(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_1214
tff(fact_9395_ATP_Olambda__1215,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,filter(C)),Uua: fun(B,filter(D)),Uub: A,Uuc: B] : aa(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_asa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = prod_filter(C,D,aa(A,filter(C),Uu,Uub),aa(B,filter(D),Uua,Uuc)) ).

% ATP.lambda_1215
tff(fact_9396_ATP_Olambda__1216,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_ave(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).

% ATP.lambda_1216
tff(fact_9397_ATP_Olambda__1217,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_arx(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(B,$o,Uua,Uuc) ) ) ).

% ATP.lambda_1217
tff(fact_9398_ATP_Olambda__1218,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_hq(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_cj(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1218
tff(fact_9399_ATP_Olambda__1219,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_ck(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_cj(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1219
tff(fact_9400_ATP_Olambda__1220,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_zr(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),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_1220
tff(fact_9401_ATP_Olambda__1221,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_zp(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),aa(num,num,bit0,one2)))))) ) ).

% ATP.lambda_1221
tff(fact_9402_ATP_Olambda__1222,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_yo(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_1222
tff(fact_9403_ATP_Olambda__1223,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_yq(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_1223
tff(fact_9404_ATP_Olambda__1224,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_zd(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_1224
tff(fact_9405_ATP_Olambda__1225,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_xq(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,aa(real,fun(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),aa(num,num,bit0,one2))))))) ) ).

% ATP.lambda_1225
tff(fact_9406_ATP_Olambda__1226,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_yz(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_1226
tff(fact_9407_ATP_Olambda__1227,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_xs(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,aa(real,fun(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),aa(num,num,bit0,one2)))))))) ) ).

% ATP.lambda_1227
tff(fact_9408_ATP_Olambda__1228,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_ahj(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,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(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_1228
tff(fact_9409_ATP_Olambda__1229,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_ahq(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,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))) ) ).

% ATP.lambda_1229
tff(fact_9410_ATP_Olambda__1230,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: set(A),Uuc: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_ale(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( ( Uuc != Uua )
           => ( aa(set(A),$o,member(A,Uuc),Uub)
             => aa(A,$o,Uu,Uuc) ) ) ) ) ).

% ATP.lambda_1230
tff(fact_9411_ATP_Olambda__1231,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_and(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),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_1231
tff(fact_9412_ATP_Olambda__1232,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_ahu(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,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_yc(A,A)))))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_yc(A,A))))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_yc(A,A)))))) ) ).

% ATP.lambda_1232
tff(fact_9413_ATP_Olambda__1233,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_ahr(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,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))) ) ).

% ATP.lambda_1233
tff(fact_9414_ATP_Olambda__1234,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_ahs(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,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))) ) ).

% ATP.lambda_1234
tff(fact_9415_ATP_Olambda__1235,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_ps(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_nb(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_1235
tff(fact_9416_ATP_Olambda__1236,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,fun(A,$o)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_ayz(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [A5: A] :
            ( ( Uub = A5 )
            & ( Uuc = A5 ) )
        | ? [A5: A,B5: A,C4: A] :
            ( ( Uub = A5 )
            & ( Uuc = C4 )
            & aa(A,$o,aa(A,fun(A,$o),Uua,A5),B5)
            & aa(A,$o,aa(A,fun(A,$o),Uu,B5),C4) ) ) ) ).

% ATP.lambda_1236
tff(fact_9417_ATP_Olambda__1237,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,fun(A,$o)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_aza(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [A5: A,B5: A] :
            ( ( Uub = A5 )
            & ( Uuc = B5 )
            & aa(A,$o,aa(A,fun(A,$o),Uu,A5),B5) )
        | ? [A5: A,B5: A,C4: A] :
            ( ( Uub = A5 )
            & ( Uuc = C4 )
            & aa(A,$o,aa(A,fun(A,$o),Uua,A5),B5)
            & aa(A,$o,aa(A,fun(A,$o),Uu,B5),C4) ) ) ) ).

% ATP.lambda_1237
tff(fact_9418_ATP_Olambda__1238,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(list(A),fun(list(A),$o)),Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_ayy(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ? [Y3: A,Ys4: list(A)] :
            ( ( Uub = nil(A) )
            & ( Uuc = cons(A,Y3,Ys4) ) )
        | ? [X: A,Y3: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = cons(A,X,Xs3) )
            & ( Uuc = cons(A,Y3,Ys4) )
            & aa(A,$o,aa(A,fun(A,$o),Uu,X),Y3) )
        | ? [X: A,Y3: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = cons(A,X,Xs3) )
            & ( Uuc = cons(A,Y3,Ys4) )
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu,X),Y3)
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu,Y3),X)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uua,Xs3),Ys4) ) ) ) ).

% ATP.lambda_1238
tff(fact_9419_ATP_Olambda__1239,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_wl(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = cons(A,Uu,list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_1239
tff(fact_9420_ATP_Olambda__1240,axiom,
    ! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bm($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1240
tff(fact_9421_ATP_Olambda__1241,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_mz(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_mx(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1241
tff(fact_9422_ATP_Olambda__1242,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_nd(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_nb(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1242
tff(fact_9423_ATP_Olambda__1243,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_my(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_mx(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1243
tff(fact_9424_ATP_Olambda__1244,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_nc(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_nb(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1244
tff(fact_9425_ATP_Olambda__1245,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_akd(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fh(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uua)))) ) ) ).

% ATP.lambda_1245
tff(fact_9426_ATP_Olambda__1246,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(B,A),Uua: fun(C,fun(D,B)),Uub: C,Uuc: D] : aa(D,A,aa(C,fun(D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_bd(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).

% ATP.lambda_1246
tff(fact_9427_ATP_Olambda__1247,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_of(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(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_1247
tff(fact_9428_ATP_Olambda__1248,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_od(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(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_1248
tff(fact_9429_ATP_Olambda__1249,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_ic(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1249
tff(fact_9430_ATP_Olambda__1250,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_cy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1250
tff(fact_9431_ATP_Olambda__1251,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo7287701948861334536_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(product_prod(B,B),$o),Uuc: A] :
          ( aa(A,$o,aa(fun(product_prod(B,B),$o),fun(A,$o),aa(B,fun(fun(product_prod(B,B),$o),fun(A,$o)),aTP_Lamp_arh(fun(A,B),fun(B,fun(fun(product_prod(B,B),$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(product_prod(B,B),$o,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uu,Uuc)),Uua)) ) ) ).

% ATP.lambda_1251
tff(fact_9432_ATP_Olambda__1252,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_baq(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_bap(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1252
tff(fact_9433_ATP_Olambda__1253,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_asp(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_aso(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1253
tff(fact_9434_ATP_Olambda__1254,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_ub(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_ua(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1254
tff(fact_9435_ATP_Olambda__1255,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_tp(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_to(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1255
tff(fact_9436_ATP_Olambda__1256,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_aws(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_awr(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1256
tff(fact_9437_ATP_Olambda__1257,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_awp(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_awo(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1257
tff(fact_9438_ATP_Olambda__1258,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: set(B),Uua: fun(A,filter(C)),Uub: fun(B,filter(D)),Uuc: A] : aa(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_asb(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(B),set(filter(product_prod(C,D))),image(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_asa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).

% ATP.lambda_1258
tff(fact_9439_ATP_Olambda__1259,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_ns(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),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_1259
tff(fact_9440_ATP_Olambda__1260,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_nu(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_1260
tff(fact_9441_ATP_Olambda__1261,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_yv(fun(B,A),fun(B,fun(fun(B,A),fun(B,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(B,A,Uu,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)))) ) ).

% ATP.lambda_1261
tff(fact_9442_ATP_Olambda__1262,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_baz(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),cons(A,Uu,Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).

% ATP.lambda_1262
tff(fact_9443_ATP_Olambda__1263,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_ny(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_1263
tff(fact_9444_ATP_Olambda__1264,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_nw(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_1264
tff(fact_9445_ATP_Olambda__1265,axiom,
    ! [C: $tType,A: $tType,B: $tType,E3: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E3,set(A))))),Uua: set(product_prod(D,E3)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(set(product_prod(D,E3)),fun(B,fun(C,set(A))),aTP_Lamp_sw(fun(B,fun(C,fun(D,fun(E3,set(A))))),fun(set(product_prod(D,E3)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D,E3)),set(set(A)),image(product_prod(D,E3),set(A),aa(fun(D,fun(E3,set(A))),fun(product_prod(D,E3),set(A)),product_case_prod(D,E3,set(A)),aa(C,fun(D,fun(E3,set(A))),aa(B,fun(C,fun(D,fun(E3,set(A)))),Uu,Uub),Uuc))),Uua)) ).

% ATP.lambda_1265
tff(fact_9446_ATP_Olambda__1266,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_anx(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X: A] :
          ( ( aa(set(A),$o,member(A,X),Uu)
           => aa(set(B),$o,member(B,aa(A,B,Uuc,X)),Uua) )
          & ( ~ aa(set(A),$o,member(A,X),Uu)
           => ( aa(A,B,Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_1266
tff(fact_9447_ATP_Olambda__1267,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_aym(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [Y3: C] :
          ( aa(set(product_prod(A,C)),$o,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y3)),Uu)
          & aa(set(product_prod(C,B)),$o,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Uuc)),Uua) ) ) ).

% ATP.lambda_1267
tff(fact_9448_ATP_Olambda__1268,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(C,B),fun(product_prod(A,B),$o),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o)),aTP_Lamp_bbd(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ? [A5: C] :
          ( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A5)),aa(C,B,Uub,A5)) )
          & aa(set(C),$o,member(C,A5),Uu) ) ) ).

% ATP.lambda_1268
tff(fact_9449_ATP_Olambda__1269,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( aa(C,$o,aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_azi(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [X: A,Y3: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X),Y3) )
          & aa(A,$o,Uu,X)
          & aa(B,$o,Uua,Y3) ) ) ).

% ATP.lambda_1269
tff(fact_9450_ATP_Olambda__1270,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] :
      aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_axv(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(C,B)),set(product_prod(C,B)),insert(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1270
tff(fact_9451_ATP_Olambda__1271,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] :
      aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_axt(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,C)),set(product_prod(A,C)),insert(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_1271
tff(fact_9452_ATP_Olambda__1272,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_ym(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_1272
tff(fact_9453_ATP_Olambda__1273,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B] : aa(B,C,aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_zj(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_zi(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_1273
tff(fact_9454_ATP_Olambda__1274,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_kf(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_ke(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1274
tff(fact_9455_ATP_Olambda__1275,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_vn($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu)),(Uua)),(Uub)),Uuc),(Uud)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_vm($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu)),(Uua)),(Uub)),(Uud))),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_1275
tff(fact_9456_ATP_Olambda__1276,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_xl(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,aa(real,fun(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_xk(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(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,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uuc)))))) ).

% ATP.lambda_1276
tff(fact_9457_ATP_Olambda__1277,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_kj(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_ki(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_1277
tff(fact_9458_ATP_Olambda__1278,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_li(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),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,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1278
tff(fact_9459_ATP_Olambda__1279,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_le(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,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1279
tff(fact_9460_ATP_Olambda__1280,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_axc(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_trancl(A,Uub))
          | ( Uuc = Uu ) )
        & ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_trancl(A,Uub))
          | ( Uud = Uua ) ) ) ) ).

% ATP.lambda_1280
tff(fact_9461_ATP_Olambda__1281,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_lf(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,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).

% ATP.lambda_1281
tff(fact_9462_ATP_Olambda__1282,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_axb(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_rtrancl(A,Uub))
        & aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_rtrancl(A,Uub)) ) ) ).

% ATP.lambda_1282
tff(fact_9463_ATP_Olambda__1283,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_ke(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_1283
tff(fact_9464_ATP_Olambda__1284,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(C) )
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(A,C),Uuc: C,Uud: A] :
          ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_amp(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,Uub,Uud),Uuc)),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uud),Uua)) ) ) ).

% ATP.lambda_1284
tff(fact_9465_ATP_Olambda__1285,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(B,C),Uuc: C,Uud: A] :
          ( aa(A,$o,aa(C,fun(A,$o),aa(fun(B,C),fun(C,fun(A,$o)),aa(B,fun(fun(B,C),fun(C,fun(A,$o))),aTP_Lamp_amy(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ( ( aa(A,B,Uu,Uud) = Uua )
           => ( aa(B,C,Uub,Uua) = Uuc ) ) ) ) ).

% ATP.lambda_1285
tff(fact_9466_ATP_Olambda__1286,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_zb(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,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),one_one(nat)))) ) ).

% ATP.lambda_1286
tff(fact_9467_ATP_Olambda__1287,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo7287701948861334536_space(A)
        & topolo7287701948861334536_space(B) )
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(product_prod(B,B),$o),Uuc: A,Uud: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aa(fun(product_prod(B,B),$o),fun(A,fun(A,$o)),aa(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o))),aTP_Lamp_ath(set(A),fun(fun(A,B),fun(fun(product_prod(B,B),$o),fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ( aa(set(A),$o,member(A,Uuc),Uu)
           => ( aa(set(A),$o,member(A,Uud),Uu)
             => aa(product_prod(B,B),$o,Uub,aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uuc)),aa(A,B,Uua,Uud))) ) ) ) ) ).

% ATP.lambda_1287
tff(fact_9468_ATP_Olambda__1288,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: fun(product_prod(A,A),$o),Uua: A,Uub: A,Uuc: A,Uud: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_arv(fun(product_prod(A,A),$o),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ( ( Uub = Uuc )
           => aa(product_prod(A,A),$o,Uu,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)) ) ) ) ).

% ATP.lambda_1288
tff(fact_9469_ATP_Olambda__1289,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: int,Uud: B] : aa(B,A,aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_aud(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uub,Uud)),aa(A,A,aa(A,fun(A,A),times_times(A),ring_1_of_int(A,Uuc)),power_int(A,aa(B,A,Uu,Uua),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uuc),one_one(int))))) ) ).

% ATP.lambda_1289
tff(fact_9470_ATP_Olambda__1290,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_vm($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_vl($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uue))),code_bit_cut_integer(Uud)) ).

% ATP.lambda_1290
tff(fact_9471_ATP_Olambda__1291,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B,Uue: A] : aa(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_zi(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,aa(A,fun(B,C),Uub,Uue),Uud)),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_zg(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),aa(set(A),set(A),insert(A,Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_1291
tff(fact_9472_ATP_Olambda__1292,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_ys(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uuc,Uub)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1292
tff(fact_9473_ATP_Olambda__1293,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_bap(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( aa(set(product_prod(A,A)),$o,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),Uu)
        | ( ( Uub = Uud )
          & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),Uua) ) ) ) ).

% ATP.lambda_1293
tff(fact_9474_ATP_Olambda__1294,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_zl(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_1294
tff(fact_9475_ATP_Olambda__1295,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_xv(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1295
tff(fact_9476_ATP_Olambda__1296,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_yk(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1296
tff(fact_9477_ATP_Olambda__1297,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_zf(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))),aa(A,B,Uud,Uue))),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1297
tff(fact_9478_ATP_Olambda__1298,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_aso(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa(A,$o,Uu,Uud)
        & aa(set(product_prod(B,B)),$o,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),aa(A,set(product_prod(B,B)),Uua,Uud)) ) ) ).

% ATP.lambda_1298
tff(fact_9479_ATP_Olambda__1299,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_vl($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_vk($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf))),code_bit_cut_integer(Uue)) ).

% ATP.lambda_1299
tff(fact_9480_ATP_Olambda__1300,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_ayi(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uue),Uuf)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuf),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uud))
           => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I4)),X_1)
            <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Uub),I4) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( aa(set(vEBT_VEBT),$o,member(vEBT_VEBT,X),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),X)
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_1300
tff(fact_9481_ATP_Olambda__1301,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_vj($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_1301
tff(fact_9482_ATP_Olambda__1302,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_vk($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = aa(product_prod(code_integer,$o),char,aa(fun(code_integer,fun($o,char)),fun(product_prod(code_integer,$o),char),product_case_prod(code_integer,$o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_vj($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug))),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_1302
tff(fact_9483_ATP_Olambda__1303,axiom,
    ! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_uu($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_1303
tff(fact_9484_ATP_Olambda__1304,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: $o,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ahx($o,fun(A,$o),(Uu)),Uua)
        <=> (Uu) ) ) ).

% ATP.lambda_1304
tff(fact_9485_ATP_Olambda__1305,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_tc($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1305
tff(fact_9486_ATP_Olambda__1306,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_avf(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1306
tff(fact_9487_ATP_Olambda__1307,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_avk(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1307
tff(fact_9488_ATP_Olambda__1308,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_axp(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1308
tff(fact_9489_ATP_Olambda__1309,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_aux(set(B),fun(A,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_1309
tff(fact_9490_ATP_Olambda__1310,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_aun(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1310
tff(fact_9491_ATP_Olambda__1311,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_avb(set(A),fun(B,set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1311
tff(fact_9492_ATP_Olambda__1312,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_avt(set(A),fun(A,set(A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1312
tff(fact_9493_ATP_Olambda__1313,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_auk(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1313
tff(fact_9494_ATP_Olambda__1314,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_rz(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1314
tff(fact_9495_ATP_Olambda__1315,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_avu(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1315
tff(fact_9496_ATP_Olambda__1316,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,fun(B,$o),aTP_Lamp_arr(fun(B,$o),fun(A,fun(B,$o)),Uu),Uua) = Uu ).

% ATP.lambda_1316
tff(fact_9497_ATP_Olambda__1317,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_axy(fun(B,B),fun(A,fun(B,B)),Uu),Uua) = Uu ).

% ATP.lambda_1317
tff(fact_9498_ATP_Olambda__1318,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_rs(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1318
tff(fact_9499_ATP_Olambda__1319,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_rd(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1319
tff(fact_9500_ATP_Olambda__1320,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_xw(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1320
tff(fact_9501_ATP_Olambda__1321,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_apd(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1321
tff(fact_9502_ATP_Olambda__1322,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_aau(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1322
tff(fact_9503_ATP_Olambda__1323,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ace(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1323
tff(fact_9504_ATP_Olambda__1324,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_rc(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1324
tff(fact_9505_ATP_Olambda__1325,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo7287701948861334536_space(B)
        & topolo7287701948861334536_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_atr(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1325
tff(fact_9506_ATP_Olambda__1326,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ano(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1326
tff(fact_9507_ATP_Olambda__1327,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_aey(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1327
tff(fact_9508_ATP_Olambda__1328,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_afa(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1328
tff(fact_9509_ATP_Olambda__1329,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_mo(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1329
tff(fact_9510_ATP_Olambda__1330,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_mv(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_1330
tff(fact_9511_ATP_Olambda__1331,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ut(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1331
tff(fact_9512_ATP_Olambda__1332,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ev(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1332
tff(fact_9513_ATP_Olambda__1333,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wk(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1333
tff(fact_9514_ATP_Olambda__1334,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_so(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1334
tff(fact_9515_ATP_Olambda__1335,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_pm(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1335
tff(fact_9516_ATP_Olambda__1336,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_afr(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1336
tff(fact_9517_ATP_Olambda__1337,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aew(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1337
tff(fact_9518_ATP_Olambda__1338,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_pc(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1338
tff(fact_9519_ATP_Olambda__1339,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_pd(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1339
tff(fact_9520_ATP_Olambda__1340,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aoe(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1340
tff(fact_9521_ATP_Olambda__1341,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aev(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1341
tff(fact_9522_ATP_Olambda__1342,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_mw(A,fun(B,A)),Uu),Uua) = Uu ).

% ATP.lambda_1342
tff(fact_9523_ATP_Olambda__1343,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_qd(B,fun(A,A)),Uu),Uua) = Uua ).

% ATP.lambda_1343
tff(fact_9524_ATP_Olambda__1344,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_bn(A,fun(B,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1344
tff(fact_9525_ATP_Olambda__1345,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_cz(complex,complex),Uu) = Uu ).

% ATP.lambda_1345
tff(fact_9526_ATP_Olambda__1346,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_df(nat,nat),Uu) = Uu ).

% ATP.lambda_1346
tff(fact_9527_ATP_Olambda__1347,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_br(int,int),Uu) = Uu ).

% ATP.lambda_1347
tff(fact_9528_ATP_Olambda__1348,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_atc(B,B),Uu) = Uu ) ).

% ATP.lambda_1348
tff(fact_9529_ATP_Olambda__1349,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_atb(B,B),Uu) = Uu ) ).

% ATP.lambda_1349
tff(fact_9530_ATP_Olambda__1350,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_yc(A,A),Uu) = Uu ) ).

% ATP.lambda_1350
tff(fact_9531_ATP_Olambda__1351,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_acf(A,A),Uu) = Uu ) ).

% ATP.lambda_1351
tff(fact_9532_ATP_Olambda__1352,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_wa(A,A),Uu) = Uu ) ).

% ATP.lambda_1352
tff(fact_9533_ATP_Olambda__1353,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_atq(A,A),Uu) = Uu ) ).

% ATP.lambda_1353
tff(fact_9534_ATP_Olambda__1354,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aac(A,A),Uu) = Uu ) ).

% ATP.lambda_1354
tff(fact_9535_ATP_Olambda__1355,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aia(A,A),Uu) = Uu ) ).

% ATP.lambda_1355
tff(fact_9536_ATP_Olambda__1356,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rv(A,A),Uu) = Uu ) ).

% ATP.lambda_1356
tff(fact_9537_ATP_Olambda__1357,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rg(A,A),Uu) = Uu ) ).

% ATP.lambda_1357
tff(fact_9538_ATP_Olambda__1358,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_asy(A,A),Uu) = Uu ) ).

% ATP.lambda_1358
tff(fact_9539_ATP_Olambda__1359,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ad(A,A),Uu) = Uu ) ).

% ATP.lambda_1359
tff(fact_9540_ATP_Olambda__1360,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ml(A,A),Uu) = Uu ).

% ATP.lambda_1360
tff(fact_9541_ATP_Olambda__1361,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_te(B,A),Uu) = top_top(A) ) ).

% ATP.lambda_1361
tff(fact_9542_ATP_Olambda__1362,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_td(B,A),Uu) = top_top(A) ) ).

% ATP.lambda_1362
tff(fact_9543_ATP_Olambda__1363,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_aum(A,set(B)),Uu) = top_top(set(B)) ).

% ATP.lambda_1363
tff(fact_9544_ATP_Olambda__1364,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_ux(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1364
tff(fact_9545_ATP_Olambda__1365,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_sk(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_1365
tff(fact_9546_ATP_Olambda__1366,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_ry(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_1366
tff(fact_9547_ATP_Olambda__1367,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_rx(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_1367
tff(fact_9548_ATP_Olambda__1368,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_uy(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_1368
tff(fact_9549_ATP_Olambda__1369,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_aup(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1369
tff(fact_9550_ATP_Olambda__1370,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_el(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1370
tff(fact_9551_ATP_Olambda__1371,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_eg(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1371
tff(fact_9552_ATP_Olambda__1372,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_bt(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1372
tff(fact_9553_ATP_Olambda__1373,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_xx(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1373
tff(fact_9554_ATP_Olambda__1374,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ae(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1374
tff(fact_9555_ATP_Olambda__1375,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_nh(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1375
tff(fact_9556_ATP_Olambda__1376,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_hc(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_1376
tff(fact_9557_ATP_Olambda__1377,axiom,
    ! [A: $tType,Uu: A] : aa(A,real,aTP_Lamp_po(A,real),Uu) = one_one(real) ).

% ATP.lambda_1377
tff(fact_9558_ATP_Olambda__1378,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_pj(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_1378
tff(fact_9559_ATP_Olambda__1379,axiom,
    ! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_bbf(A,option(C)),Uu) = none(C) ).

% ATP.lambda_1379
tff(fact_9560_ATP_Olambda__1380,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_ly(real,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1380
tff(fact_9561_ATP_Olambda__1381,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_pg(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1381
tff(fact_9562_ATP_Olambda__1382,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_ag(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1382
tff(fact_9563_ATP_Olambda__1383,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_ph(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1383
tff(fact_9564_ATP_Olambda__1384,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_tn(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1384
tff(fact_9565_ATP_Olambda__1385,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_axs(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1385

% Type constructors (829)
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_1,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_2,axiom,
    ! [A15: $tType] : bounded_lattice(filter(A15)) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_3,axiom,
    bounded_lattice($o) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_4,axiom,
    ! [A15: $tType] : bounded_lattice(set(A15)) ).

tff(tcon_fun___Lattices_Obounded__lattice_5,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bounded_lattice(A16)
     => bounded_lattice(fun(A15,A16)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => condit1219197933456340205attice(fun(A15,A16)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( counta3822494911875563373attice(A16)
     => counta3822494911875563373attice(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple592849572758109894attice(A16)
     => comple592849572758109894attice(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple489889107523837845lgebra(A16)
     => comple489889107523837845lgebra(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bounded_lattice(A16)
     => bounde4967611905675639751up_bot(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A15: $tType,A16: $tType] :
      ( bounded_lattice(A16)
     => bounde4346867609351753570nf_top(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( comple6319245703460814977attice(A16)
     => comple6319245703460814977attice(fun(A15,A16)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A15: $tType,A16: $tType] :
      ( boolea8198339166811842893lgebra(A16)
     => boolea8198339166811842893lgebra(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A15: $tType,A16: $tType] :
      ( semilattice_sup(A16)
     => semilattice_sup(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A15: $tType,A16: $tType] :
      ( semilattice_inf(A16)
     => semilattice_inf(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( distrib_lattice(A16)
     => distrib_lattice(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_OSup,axiom,
    ! [A15: $tType,A16: $tType] :
      ( complete_Sup(A16)
     => complete_Sup(fun(A15,A16)) ) ).

tff(tcon_fun___Complete__Lattices_OInf,axiom,
    ! [A15: $tType,A16: $tType] :
      ( complete_Inf(A16)
     => complete_Inf(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A15: $tType,A16: $tType] :
      ( order_top(A16)
     => order_top(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A15: $tType,A16: $tType] :
      ( order_bot(A16)
     => order_bot(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A15: $tType,A16: $tType] :
      ( preorder(A16)
     => preorder(fun(A15,A16)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( finite_finite(A15)
        & finite_finite(A16) )
     => finite_finite(fun(A15,A16)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A15: $tType,A16: $tType] :
      ( lattice(A16)
     => lattice(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A15: $tType,A16: $tType] :
      ( order(A16)
     => order(fun(A15,A16)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ord(A16)
     => ord(fun(A15,A16)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A15: $tType,A16: $tType] :
      ( uminus(A16)
     => uminus(fun(A15,A16)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A15: $tType,A16: $tType] :
      ( minus(A16)
     => minus(fun(A15,A16)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_6,axiom,
    condit1219197933456340205attice(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___Limits_Otopological__comm__monoid__mult,axiom,
    topolo4987421752381908075d_mult(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___GCD_Osemiring__gcd__mult__normalize,axiom,
    semiri6843258321239162965malize(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___Rings_Oordered__semiring__0,axiom,
    ordered_semiring_0(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__sup_7,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_8,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_9,axiom,
    distrib_lattice(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___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___Complete__Lattices_OSup_10,axiom,
    complete_Sup(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OInf_11,axiom,
    complete_Inf(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_12,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___Lattices_Olattice_13,axiom,
    lattice(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___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_14,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___Orderings_Oord_15,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_16,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___Groups_Ominus_17,axiom,
    minus(int) ).

tff(tcon_Int_Oint___GCD_Oring__gcd,axiom,
    ring_gcd(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_18,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_19,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_20,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_21,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_22,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_23,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_24,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_25,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_26,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_27,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_28,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_29,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_30,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_31,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_32,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_33,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_34,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_35,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_36,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_37,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_38,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_39,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_40,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_41,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_42,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_43,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_44,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_45,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_46,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_47,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_48,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_49,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_50,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_51,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_52,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_53,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_54,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_55,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_56,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_57,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_58,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_59,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_60,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_61,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_62,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_63,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_64,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_65,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_66,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_67,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_68,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_69,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_70,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_71,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_72,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_73,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_74,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_75,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_76,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_77,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_78,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OSup_79,axiom,
    complete_Sup(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OInf_80,axiom,
    complete_Inf(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_81,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_82,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_83,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_84,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_85,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_86,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_87,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_88,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_89,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_90,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_91,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_92,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_93,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_94,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_95,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_96,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_97,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_98,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_99,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_100,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_101,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_102,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_103,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_104,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_105,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_106,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_107,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_108,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_109,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_110,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_111,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_112,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_113,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_114,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_115,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_116,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_117,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_118,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_119,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_120,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_121,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_122,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_123,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_124,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_125,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_126,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_127,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_128,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_129,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_130,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_131,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_132,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_133,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_134,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_135,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_136,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_137,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_138,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_139,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_140,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_141,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_142,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_143,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_144,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_145,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_146,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_147,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_148,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_149,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_150,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_151,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_152,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_153,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_154,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_155,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_156,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_157,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_158,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_159,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_160,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_161,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_162,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_163,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_164,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_165,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_166,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_167,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_168,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_169,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_170,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_171,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_172,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_173,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_174,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_175,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_176,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_177,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_178,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_179,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_180,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_181,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_182,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_183,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_184,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_185,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_186,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_187,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_188,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_189,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_190,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_191,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_192,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_193,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_194,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_195,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_196,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_197,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_198,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_199,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_200,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_201,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_202,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_203,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_204,axiom,
    ! [A15: $tType] : condit1219197933456340205attice(set(A15)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_205,axiom,
    ! [A15: $tType] : counta3822494911875563373attice(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_206,axiom,
    ! [A15: $tType] : comple592849572758109894attice(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_207,axiom,
    ! [A15: $tType] : comple489889107523837845lgebra(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_208,axiom,
    ! [A15: $tType] : bounde4967611905675639751up_bot(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_209,axiom,
    ! [A15: $tType] : bounde4346867609351753570nf_top(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_210,axiom,
    ! [A15: $tType] : comple6319245703460814977attice(set(A15)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_211,axiom,
    ! [A15: $tType] : boolea8198339166811842893lgebra(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_212,axiom,
    ! [A15: $tType] : semilattice_sup(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_213,axiom,
    ! [A15: $tType] : semilattice_inf(set(A15)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_214,axiom,
    ! [A15: $tType] : distrib_lattice(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_OSup_215,axiom,
    ! [A15: $tType] : complete_Sup(set(A15)) ).

tff(tcon_Set_Oset___Complete__Lattices_OInf_216,axiom,
    ! [A15: $tType] : complete_Inf(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_217,axiom,
    ! [A15: $tType] : order_top(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_218,axiom,
    ! [A15: $tType] : order_bot(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_219,axiom,
    ! [A15: $tType] : preorder(set(A15)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_220,axiom,
    ! [A15: $tType] :
      ( finite_finite(A15)
     => finite_finite(set(A15)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_221,axiom,
    ! [A15: $tType] : lattice(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oorder_222,axiom,
    ! [A15: $tType] : order(set(A15)) ).

tff(tcon_Set_Oset___Orderings_Oord_223,axiom,
    ! [A15: $tType] : ord(set(A15)) ).

tff(tcon_Set_Oset___Groups_Ouminus_224,axiom,
    ! [A15: $tType] : uminus(set(A15)) ).

tff(tcon_Set_Oset___Groups_Ominus_225,axiom,
    ! [A15: $tType] : minus(set(A15)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_226,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_227,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_228,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_229,axiom,
    comple489889107523837845lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_230,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_231,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_232,axiom,
    topolo8865339358273720382pology($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_233,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_234,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_235,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_236,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_237,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_238,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_239,axiom,
    topological_t1_space($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_240,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_241,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_242,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OSup_243,axiom,
    complete_Sup($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_OInf_244,axiom,
    complete_Inf($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_245,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_246,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_247,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_248,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_249,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_250,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_251,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_252,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_253,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_254,axiom,
    minus($o) ).

tff(tcon_List_Olist___Nat_Osize_255,axiom,
    ! [A15: $tType] : size(list(A15)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_256,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_257,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_258,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
    condit5016429287641298734tinuum(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_259,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_260,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_261,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_262,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_263,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_264,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_265,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_266,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_267,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_268,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_269,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oopen__uniformity,axiom,
    topolo569519726778239578ormity(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_270,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_271,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_272,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_273,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_274,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_275,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_276,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_277,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_278,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_279,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_280,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_281,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_282,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_283,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_284,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_285,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_286,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_287,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_288,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_289,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_290,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Odist__norm,axiom,
    real_V6936659425649961206t_norm(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_291,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_292,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_293,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_294,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_295,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_296,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_297,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_298,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_299,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_300,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_301,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_302,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_303,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_304,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_305,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_306,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_307,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_308,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_309,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_310,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_311,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_312,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_313,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_314,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_315,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_316,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_317,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Complete__Lattices_OSup_318,axiom,
    complete_Sup(real) ).

tff(tcon_Real_Oreal___Complete__Lattices_OInf_319,axiom,
    complete_Inf(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_320,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_321,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_322,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_323,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_324,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_325,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_326,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_327,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_328,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_329,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_330,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_331,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_332,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_333,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_334,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_335,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_336,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_337,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_338,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_339,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_340,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_341,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_342,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_343,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_344,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_345,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_346,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_347,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_348,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_349,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_350,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_351,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_352,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_353,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_354,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_355,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_356,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_357,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_358,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_359,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_360,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_361,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_362,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_363,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_364,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_365,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_366,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Nat_Osize_367,axiom,
    size(char) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_368,axiom,
    ! [A15: $tType] : condit1219197933456340205attice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_369,axiom,
    ! [A15: $tType] : counta3822494911875563373attice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_370,axiom,
    ! [A15: $tType] : bounde4967611905675639751up_bot(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_371,axiom,
    ! [A15: $tType] : bounde4346867609351753570nf_top(filter(A15)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_372,axiom,
    ! [A15: $tType] : comple6319245703460814977attice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_373,axiom,
    ! [A15: $tType] : semilattice_sup(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_374,axiom,
    ! [A15: $tType] : semilattice_inf(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_375,axiom,
    ! [A15: $tType] : distrib_lattice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OSup_376,axiom,
    ! [A15: $tType] : complete_Sup(filter(A15)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OInf_377,axiom,
    ! [A15: $tType] : complete_Inf(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_378,axiom,
    ! [A15: $tType] : order_top(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_379,axiom,
    ! [A15: $tType] : order_bot(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_380,axiom,
    ! [A15: $tType] : preorder(filter(A15)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_381,axiom,
    ! [A15: $tType] : lattice(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_382,axiom,
    ! [A15: $tType] : order(filter(A15)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_383,axiom,
    ! [A15: $tType] : ord(filter(A15)) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_384,axiom,
    ! [A15: $tType] :
      ( finite_finite(A15)
     => finite_finite(option(A15)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_385,axiom,
    ! [A15: $tType] : size(option(A15)) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_386,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_387,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_388,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_389,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_390,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_391,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_392,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_393,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_394,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_395,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Oopen__uniformity_396,axiom,
    topolo569519726778239578ormity(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_397,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_398,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_399,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_400,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_401,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_402,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_403,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_404,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_405,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_406,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_407,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_408,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_409,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_410,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_411,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_412,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_413,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_414,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_415,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Odist__norm_416,axiom,
    real_V6936659425649961206t_norm(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_417,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_418,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_419,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_420,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_421,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_422,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_423,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_424,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_425,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_426,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_427,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_428,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_429,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_430,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_431,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_432,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_433,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_434,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_435,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_436,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_437,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_438,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_439,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_440,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_441,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_442,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_443,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_444,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_445,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_446,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_447,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_448,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_449,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_450,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_451,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_452,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_453,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_454,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_455,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_456,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_457,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_458,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_459,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_460,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_461,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_462,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_463,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_464,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_465,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_466,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_467,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_468,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_469,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_470,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_471,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_472,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_473,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_474,axiom,
    bounde4346867609351753570nf_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_475,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_476,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_477,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_478,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_479,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_480,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_481,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_482,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_483,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_484,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_485,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_486,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_487,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_488,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_489,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_490,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_491,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_492,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_OSup_493,axiom,
    complete_Sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_OInf_494,axiom,
    complete_Inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_495,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_496,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_497,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_498,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_499,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_500,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_501,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_502,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_503,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_504,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_505,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_506,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_507,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_508,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_509,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_510,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_511,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_512,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_513,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_514,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_515,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_516,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_517,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_518,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_519,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_520,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_521,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_522,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( topolo4958980785337419405_space(A15)
        & topolo4958980785337419405_space(A16) )
     => topolo4958980785337419405_space(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_523,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( topological_t2_space(A15)
        & topological_t2_space(A16) )
     => topological_t2_space(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_524,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( topological_t1_space(A15)
        & topological_t1_space(A16) )
     => topological_t1_space(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_525,axiom,
    ! [A15: $tType,A16: $tType] :
      ( ( finite_finite(A15)
        & finite_finite(A16) )
     => finite_finite(product_prod(A15,A16)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_526,axiom,
    ! [A15: $tType,A16: $tType] : size(product_prod(A15,A16)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_527,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_528,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_529,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_530,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_531,axiom,
    comple489889107523837845lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_532,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_533,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_534,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_535,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_536,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_537,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_538,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_539,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OSup_540,axiom,
    complete_Sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OInf_541,axiom,
    complete_Inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_542,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_543,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_544,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_545,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_546,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_547,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_548,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_549,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_550,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_551,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_552,axiom,
    minus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_553,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_554,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_555,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_556,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_557,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_558,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_559,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_560,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_561,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_562,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_563,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_564,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_565,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_566,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_567,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_568,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_569,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_570,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_571,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_572,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_573,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_574,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_575,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_576,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_577,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_578,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_579,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_580,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_581,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_582,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_583,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_584,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_585,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_586,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_587,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_588,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_589,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_590,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_591,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_592,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_593,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_594,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_595,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_596,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_597,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_598,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_599,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_600,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_601,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_602,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_603,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_604,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_605,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_606,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_607,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_608,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_609,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_610,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_611,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_612,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_613,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_614,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_615,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_616,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_617,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_618,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_619,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_620,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_621,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_622,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_623,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_624,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_625,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_626,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_627,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_628,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_629,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_630,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_631,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_632,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_633,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_634,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_635,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_636,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_637,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_638,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_639,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_640,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_641,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_642,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_643,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_644,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_645,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_646,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_647,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_648,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_649,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_650,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_651,axiom,
    dvd(code_integer) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_652,axiom,
    size(vEBT_VEBT) ).

% Helper facts (3)
tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X9: A,Y: A] :
      ( ( X9 != Y )
      | aa(A,$o,fequal(A,X9),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X9: A,Y: A] :
      ( ~ aa(A,$o,fequal(A,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X7: A] : aa(A,$o,P,X7) ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

% Conjectures (1)
tff(conj_0,conjecture,
    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)),deg,treeList,summary),xa) = 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,bit0,aa(num,num,bit1,aa(num,num,bit0,one2)))))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),deg),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

%------------------------------------------------------------------------------