%------------------------------------------------------------------------------
% File     : ITP244_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Succ 00501_031916
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0070_VEBT_Succ_00501_031916 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11392 (4279 unt;1516 typ;   0 def)
%            Number of atoms       : 18542 (8309 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 18311 (1883   ~; 337   |;1887   &)
%                                         (1984 <=>;12220  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   31 (   2 avg)
%            Number of FOOLs       :  863 ( 473 fml; 390 var)
%            Number of X terms     :  642 (   0  []; 544 ite;  98 let)
%            Number of types       :   13 (  12 usr)
%            Number of type conns  : 1285 (1088   >; 197   *;   0   +;   0  <<)
%            Number of predicates  :  230 ( 227 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1296 (1296 usr;  73 con; 0-8 aty)
%            Number of variables   : 33311 (30110   !; 845   ?;33311   :)
%                                         (2356  !>;   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 00:38:22.502
%------------------------------------------------------------------------------
% 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_Sum__Type_Osum,type,
    sum_sum: ( $tType * $tType ) > $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_fun,type,
    fun: ( $tType * $tType ) > $tType ).

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

% Explicit typings (1497)
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_Groups_Omonoid__add,type,
    monoid_add: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_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_Ot2__space,type,
    topological_t2_space: 
      !>[A: $tType] : $o ).

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

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

tff(sy_cl_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_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_Rings_Olinordered__semiring__1__strict,type,
    linord715952674999750819strict: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_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_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__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] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__aio____,type,
    aTP_Lamp_aio: 
      !>[A: $tType,B: $tType] : fun(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__aj____,type,
    aTP_Lamp_aj: 
      !>[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__al____,type,
    aTP_Lamp_al: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__kr____,type,
    aTP_Lamp_kr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ks____,type,
    aTP_Lamp_ks: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__kx____,type,
    aTP_Lamp_kx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ky____,type,
    aTP_Lamp_ky: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kz____,type,
    aTP_Lamp_kz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__la____,type,
    aTP_Lamp_la: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lb____,type,
    aTP_Lamp_lb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lc____,type,
    aTP_Lamp_lc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ld____,type,
    aTP_Lamp_ld: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__le____,type,
    aTP_Lamp_le: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lf____,type,
    aTP_Lamp_lf: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lj____,type,
    aTP_Lamp_lj: 
      !>[A: $tType,B: $tType] : fun(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] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lp____,type,
    aTP_Lamp_lp: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mc____,type,
    aTP_Lamp_mc: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mf____,type,
    aTP_Lamp_mf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mg____,type,
    aTP_Lamp_mg: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ml____,type,
    aTP_Lamp_ml: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mm____,type,
    aTP_Lamp_mm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mn____,type,
    aTP_Lamp_mn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mo____,type,
    aTP_Lamp_mo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mp____,type,
    aTP_Lamp_mp: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mt____,type,
    aTP_Lamp_mt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mu____,type,
    aTP_Lamp_mu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mv____,type,
    aTP_Lamp_mv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

tff(sy_c_ATP_058Lamp__na____,type,
    aTP_Lamp_na: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nb____,type,
    aTP_Lamp_nb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__nc____,type,
    aTP_Lamp_nc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nd____,type,
    aTP_Lamp_nd: 
      !>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__nh____,type,
    aTP_Lamp_nh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ni____,type,
    aTP_Lamp_ni: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nj____,type,
    aTP_Lamp_nj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nk____,type,
    aTP_Lamp_nk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nl____,type,
    aTP_Lamp_nl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nm____,type,
    aTP_Lamp_nm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nn____,type,
    aTP_Lamp_nn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__no____,type,
    aTP_Lamp_no: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__np____,type,
    aTP_Lamp_np: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nq____,type,
    aTP_Lamp_nq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nr____,type,
    aTP_Lamp_nr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ns____,type,
    aTP_Lamp_ns: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nt____,type,
    aTP_Lamp_nt: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ny____,type,
    aTP_Lamp_ny: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__oq____,type,
    aTP_Lamp_oq: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ov____,type,
    aTP_Lamp_ov: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ow____,type,
    aTP_Lamp_ow: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ox____,type,
    aTP_Lamp_ox: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pd____,type,
    aTP_Lamp_pd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pe____,type,
    aTP_Lamp_pe: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__pf____,type,
    aTP_Lamp_pf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pg____,type,
    aTP_Lamp_pg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__pn____,type,
    aTP_Lamp_pn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__po____,type,
    aTP_Lamp_po: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pp____,type,
    aTP_Lamp_pp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pq____,type,
    aTP_Lamp_pq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pr____,type,
    aTP_Lamp_pr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ps____,type,
    aTP_Lamp_ps: 
      !>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__px____,type,
    aTP_Lamp_px: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__pz____,type,
    aTP_Lamp_pz: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qd____,type,
    aTP_Lamp_qd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qe____,type,
    aTP_Lamp_qe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qf____,type,
    aTP_Lamp_qf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qg____,type,
    aTP_Lamp_qg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qh____,type,
    aTP_Lamp_qh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qi____,type,
    aTP_Lamp_qi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qj____,type,
    aTP_Lamp_qj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qk____,type,
    aTP_Lamp_qk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ql____,type,
    aTP_Lamp_ql: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qm____,type,
    aTP_Lamp_qm: 
      !>[A: $tType,B: $tType] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qp____,type,
    aTP_Lamp_qp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qq____,type,
    aTP_Lamp_qq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qr____,type,
    aTP_Lamp_qr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qs____,type,
    aTP_Lamp_qs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__qu____,type,
    aTP_Lamp_qu: 
      !>[A: $tType,B: $tType] : ( 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] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qz____,type,
    aTP_Lamp_qz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ra____,type,
    aTP_Lamp_ra: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__rc____,type,
    aTP_Lamp_rc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rd____,type,
    aTP_Lamp_rd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__re____,type,
    aTP_Lamp_re: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rf____,type,
    aTP_Lamp_rf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rg____,type,
    aTP_Lamp_rg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rh____,type,
    aTP_Lamp_rh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ri____,type,
    aTP_Lamp_ri: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rj____,type,
    aTP_Lamp_rj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rk____,type,
    aTP_Lamp_rk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rl____,type,
    aTP_Lamp_rl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rm____,type,
    aTP_Lamp_rm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rn____,type,
    aTP_Lamp_rn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ro____,type,
    aTP_Lamp_ro: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rp____,type,
    aTP_Lamp_rp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rq____,type,
    aTP_Lamp_rq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rr____,type,
    aTP_Lamp_rr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rs____,type,
    aTP_Lamp_rs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rt____,type,
    aTP_Lamp_rt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ru____,type,
    aTP_Lamp_ru: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rv____,type,
    aTP_Lamp_rv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rw____,type,
    aTP_Lamp_rw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rx____,type,
    aTP_Lamp_rx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__rz____,type,
    aTP_Lamp_rz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sa____,type,
    aTP_Lamp_sa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sb____,type,
    aTP_Lamp_sb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sc____,type,
    aTP_Lamp_sc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sd____,type,
    aTP_Lamp_sd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__se____,type,
    aTP_Lamp_se: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sf____,type,
    aTP_Lamp_sf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sg____,type,
    aTP_Lamp_sg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sh____,type,
    aTP_Lamp_sh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__si____,type,
    aTP_Lamp_si: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sj____,type,
    aTP_Lamp_sj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sk____,type,
    aTP_Lamp_sk: 
      !>[A: $tType,B: $tType] : 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] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__tf____,type,
    aTP_Lamp_tf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tg____,type,
    aTP_Lamp_tg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__th____,type,
    aTP_Lamp_th: 
      !>[A: $tType,B: $tType] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__to____,type,
    aTP_Lamp_to: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tp____,type,
    aTP_Lamp_tp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tq____,type,
    aTP_Lamp_tq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tr____,type,
    aTP_Lamp_tr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ts____,type,
    aTP_Lamp_ts: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tt____,type,
    aTP_Lamp_tt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tu____,type,
    aTP_Lamp_tu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tv____,type,
    aTP_Lamp_tv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tw____,type,
    aTP_Lamp_tw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tx____,type,
    aTP_Lamp_tx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ty____,type,
    aTP_Lamp_ty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tz____,type,
    aTP_Lamp_tz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ua____,type,
    aTP_Lamp_ua: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ub____,type,
    aTP_Lamp_ub: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uc____,type,
    aTP_Lamp_uc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ud____,type,
    aTP_Lamp_ud: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ue____,type,
    aTP_Lamp_ue: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uf____,type,
    aTP_Lamp_uf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ug____,type,
    aTP_Lamp_ug: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uh____,type,
    aTP_Lamp_uh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ui____,type,
    aTP_Lamp_ui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uj____,type,
    aTP_Lamp_uj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uk____,type,
    aTP_Lamp_uk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ul____,type,
    aTP_Lamp_ul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__um____,type,
    aTP_Lamp_um: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__un____,type,
    aTP_Lamp_un: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uo____,type,
    aTP_Lamp_uo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__up____,type,
    aTP_Lamp_up: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uq____,type,
    aTP_Lamp_uq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ur____,type,
    aTP_Lamp_ur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__us____,type,
    aTP_Lamp_us: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ut____,type,
    aTP_Lamp_ut: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uu____,type,
    aTP_Lamp_uu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uv____,type,
    aTP_Lamp_uv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uw____,type,
    aTP_Lamp_uw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ux____,type,
    aTP_Lamp_ux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

tff(sy_c_ATP_058Lamp__vb____,type,
    aTP_Lamp_vb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vc____,type,
    aTP_Lamp_vc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vd____,type,
    aTP_Lamp_vd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__vf____,type,
    aTP_Lamp_vf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vg____,type,
    aTP_Lamp_vg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vh____,type,
    aTP_Lamp_vh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vi____,type,
    aTP_Lamp_vi: 
      !>[A: $tType,B: $tType] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vr____,type,
    aTP_Lamp_vr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vs____,type,
    aTP_Lamp_vs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vt____,type,
    aTP_Lamp_vt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vu____,type,
    aTP_Lamp_vu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vv____,type,
    aTP_Lamp_vv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vw____,type,
    aTP_Lamp_vw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vx____,type,
    aTP_Lamp_vx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vy____,type,
    aTP_Lamp_vy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vz____,type,
    aTP_Lamp_vz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wa____,type,
    aTP_Lamp_wa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wb____,type,
    aTP_Lamp_wb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wc____,type,
    aTP_Lamp_wc: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xf____,type,
    aTP_Lamp_xf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xg____,type,
    aTP_Lamp_xg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xh____,type,
    aTP_Lamp_xh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xi____,type,
    aTP_Lamp_xi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xj____,type,
    aTP_Lamp_xj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xk____,type,
    aTP_Lamp_xk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xl____,type,
    aTP_Lamp_xl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xm____,type,
    aTP_Lamp_xm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xn____,type,
    aTP_Lamp_xn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xo____,type,
    aTP_Lamp_xo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xp____,type,
    aTP_Lamp_xp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xq____,type,
    aTP_Lamp_xq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xr____,type,
    aTP_Lamp_xr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xs____,type,
    aTP_Lamp_xs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xt____,type,
    aTP_Lamp_xt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xu____,type,
    aTP_Lamp_xu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xv____,type,
    aTP_Lamp_xv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xw____,type,
    aTP_Lamp_xw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xx____,type,
    aTP_Lamp_xx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xy____,type,
    aTP_Lamp_xy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xz____,type,
    aTP_Lamp_xz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ya____,type,
    aTP_Lamp_ya: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yb____,type,
    aTP_Lamp_yb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yc____,type,
    aTP_Lamp_yc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yd____,type,
    aTP_Lamp_yd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ye____,type,
    aTP_Lamp_ye: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yf____,type,
    aTP_Lamp_yf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yg____,type,
    aTP_Lamp_yg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yh____,type,
    aTP_Lamp_yh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yi____,type,
    aTP_Lamp_yi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yj____,type,
    aTP_Lamp_yj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yk____,type,
    aTP_Lamp_yk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yl____,type,
    aTP_Lamp_yl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ym____,type,
    aTP_Lamp_ym: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yn____,type,
    aTP_Lamp_yn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yo____,type,
    aTP_Lamp_yo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yp____,type,
    aTP_Lamp_yp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yq____,type,
    aTP_Lamp_yq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yr____,type,
    aTP_Lamp_yr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ys____,type,
    aTP_Lamp_ys: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yt____,type,
    aTP_Lamp_yt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yu____,type,
    aTP_Lamp_yu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yv____,type,
    aTP_Lamp_yv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yw____,type,
    aTP_Lamp_yw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yy____,type,
    aTP_Lamp_yy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yz____,type,
    aTP_Lamp_yz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__za____,type,
    aTP_Lamp_za: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zb____,type,
    aTP_Lamp_zb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zc____,type,
    aTP_Lamp_zc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zd____,type,
    aTP_Lamp_zd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ze____,type,
    aTP_Lamp_ze: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zk____,type,
    aTP_Lamp_zk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : fun(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__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_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 > fun(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 > fun(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_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_Oprincipal,type,
    principal: 
      !>[A: $tType] : ( set(A) > filter(A) ) ).

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_Ocomp__fun__commute__on,type,
    finite4664212375090638736ute_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite: 
      !>[A: $tType] : fun(set(A),$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_Finite__Set_Ofolding__on,type,
    finite_folding_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on_OF,type,
    finite_folding_F: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B ) > fun(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_Oid,type,
    id: 
      !>[A: $tType] : fun(A,A) ).

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_Ooverride__on,type,
    override_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * set(A) ) > fun(A,B) ) ).

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] : ( ( A * 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_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_Groups__List_Omonoid__add__class_Osum__list,type,
    groups8242544230860333062m_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
    groups5270119922927024881d_list: 
      !>[A: $tType] : ( list(A) > 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_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : 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_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] : fun(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_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(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_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[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] : fun(list(A),fun(list(A),list(A))) ).

tff(sy_c_List_Oarg__min__list,type,
    arg_min_list: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > 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_Odrop,type,
    drop: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

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

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_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( fun(A,$o) > fun(list(A),list(A)) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(A) ) ).

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

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

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

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_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(B,A),fun(B,fun(list(B),list(B)))) ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,$o)) > fun(fun(B,A),fun(set(B),list(B))) ) ).

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__key__list__of__set,type,
    linord144544945434240204of_set: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(set(B),list(B)) ) ).

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

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

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

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

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

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

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

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

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

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

tff(sy_c_List_Olistrel,type,
    listrel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).

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_Olistrelp,type,
    listrelp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

tff(sy_c_List_Olistset,type,
    listset: 
      !>[A: $tType] : ( list(set(A)) > set(list(A)) ) ).

tff(sy_c_List_Omap__filter,type,
    map_filter: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).

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

tff(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$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_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Opartition,type,
    partition: 
      !>[A: $tType] : ( fun(A,$o) > fun(list(A),product_prod(list(A),list(A))) ) ).

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

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

tff(sy_c_List_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_Oremdups__adj__rel,type,
    remdups_adj_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

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

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

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

tff(sy_c_List_Orev,type,
    rev: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_List_Orotate,type,
    rotate: 
      !>[A: $tType] : ( nat > fun(list(A),list(A)) ) ).

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : fun(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_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).

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

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

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

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

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

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

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_List_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_Map_Odom,type,
    dom: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).

tff(sy_c_Map_Ograph,type,
    graph: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Map_Omap__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__upds,type,
    map_upds: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

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: fun(list(nat),nat) ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),$o)) ).

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_Orec__option,type,
    rec_option: 
      !>[C: $tType,A: $tType] : ( ( C * fun(A,C) ) > fun(option(A),C) ) ).

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__Relation_OAbove,type,
    order_Above: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OUnder,type,
    order_Under: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OUnderS,type,
    order_UnderS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_Olinear__order__on,type,
    order_679001287576687338der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Ostrict__linear__order__on,type,
    order_5396836661320670305der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

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

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

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,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_Oord__class_Omin,type,
    ord_min: 
      !>[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_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_Product__Type_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun(product_prod(int,int),rat) ).

tff(sy_c_Rat_OFract,type,
    fract: ( int * int ) > rat ).

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

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun(rat,product_prod(int,int)) ).

tff(sy_c_Rat_Ofield__char__0__class_ORats,type,
    field_char_0_Rats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
    field_char_0_of_rat: 
      !>[A: $tType] : fun(rat,A) ).

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

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

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,$o) ).

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

tff(sy_c_Rat_Oratrel,type,
    ratrel: ( product_prod(int,int) * product_prod(int,int) ) > $o ).

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] : ( real > A ) ).

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

tff(sy_c_Relation_OField,type,
    field2: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).

tff(sy_c_Relation_OId,type,
    id2: 
      !>[A: $tType] : set(product_prod(A,A)) ).

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

tff(sy_c_Relation_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Orefl__on,type,
    refl_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

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_Relation_Ototal__on,type,
    total_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
    algebr8660921524188924756oprime: 
      !>[A: $tType] : ( ( A * A ) > $o ) ).

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

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : ( ( A * 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) * A ) > $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_Oimage,type,
    image: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > set(B) ) ).

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

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

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

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

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

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : ( A > set(A) ) ).

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

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

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

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

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

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

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

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o * $o ) > 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_Omonoseq,type,
    topological_monoseq: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
    topolo1002775350975398744n_open: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
    topolo3827282254853284352ce_Lim: 
      !>[F: $tType,A: $tType] : ( ( filter(F) * fun(F,A) ) > A ) ).

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

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
    topolo7230453075368039082e_nhds: 
      !>[A: $tType] : ( A > filter(A) ) ).

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

tff(sy_c_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] : ( 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__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__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__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_Wellfounded_Opred__nat,type,
    pred_nat: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Owf,type,
    wf: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

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_fNot,type,
    fNot: fun($o,$o) ).

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

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( ( A * 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_sxa____,type,
    sxa: nat ).

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

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (9086)
tff(fact_0__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = na ).

% \<open>deg div 2 = n\<close>
tff(fact_1__C5_Ohyps_C_I3_J,axiom,
    ! [Xb: nat,Sxb: nat] :
      ( ( vEBT_vebt_succ(summary,Xb) = aa(nat,option(nat),some(nat),Sxb) )
    <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(summary),Xb,Sxb) ) ).

% "5.hyps"(3)
tff(fact_2__C5_Ohyps_C_I9_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),mi),ma) ).

% "5.hyps"(9)
tff(fact_3_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_4_option_Oinject,axiom,
    ! [A: $tType,X2: A,Y2: A] :
      ( ( aa(A,option(A),some(A),X2) = aa(A,option(A),some(A),Y2) )
    <=> ( X2 = Y2 ) ) ).

% option.inject
tff(fact_5_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_6_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: 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),A3),B3) )
    <=> ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_7__C5_Ohyps_C_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
     => ( vEBT_invar_vebt(X,na)
        & ! [Xa: nat,Xb2: nat] :
            ( ( vEBT_vebt_succ(X,Xa) = aa(nat,option(nat),some(nat),Xb2) )
          <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(X),Xa,Xb2) ) ) ) ).

% "5.hyps"(1)
tff(fact_8_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_9__C5_Ohyps_C_I2_J,axiom,
    vEBT_invar_vebt(summary,m) ).

% "5.hyps"(2)
tff(fact_10_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_11_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod(nat,nat)] :
      ~ ! [K: nat,M: nat] : Xb != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K),M) ).

% prod_decode_aux.cases
tff(fact_12__C5_Ohyps_C_I6_J,axiom,
    deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) ).

% "5.hyps"(6)
tff(fact_13__C5_Ohyps_C_I8_J,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) ) ).

% "5.hyps"(8)
tff(fact_14_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A4: A,B4: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) ).

% old.prod.exhaust
tff(fact_15_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_16_min__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat] :
      ( vEBT_VEBT_min_in_set(Xs,Xb)
    <=> ( member(nat,Xb,Xs)
        & ! [X3: nat] :
            ( member(nat,X3,Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),X3) ) ) ) ).

% min_in_set_def
tff(fact_17_max__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat] :
      ( vEBT_VEBT_max_in_set(Xs,Xb)
    <=> ( member(nat,Xb,Xs)
        & ! [X3: nat] :
            ( member(nat,X3,Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Xb) ) ) ) ).

% max_in_set_def
tff(fact_18_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_19_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_20__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_21_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 )
       => ( ! [X: vEBT_VEBT] :
              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
             => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) )
          & ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_1) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_22_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_23_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_24_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_25_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_26__C5_Ohyps_C_I5_J,axiom,
    m = aa(nat,nat,suc,na) ).

% "5.hyps"(5)
tff(fact_27_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_28_prod__induct7,axiom,
    ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C2),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D2),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2)))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),$o,P,Xb) ) ).

% prod_induct7
tff(fact_29_prod__induct6,axiom,
    ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B4),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C2),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D2),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),$o,P,Xb) ) ).

% prod_induct6
tff(fact_30_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,Xb) ) ).

% prod_induct5
tff(fact_31_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,B4: B,C2: C,D2: D] : aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D2))))
     => aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,Xb) ) ).

% prod_induct4
tff(fact_32_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,B4: B,C2: C] : aa(product_prod(A,product_prod(B,C)),$o,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2)))
     => aa(product_prod(A,product_prod(B,C)),$o,P,Xb) ) ).

% prod_induct3
tff(fact_33_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C2),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D2),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2)))))) ).

% prod_cases7
tff(fact_34_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B4),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C2),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D2),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2))))) ).

% prod_cases6
tff(fact_35_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))) ).

% prod_cases5
tff(fact_36_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A4: A,B4: B,C2: C,D2: 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)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D2))) ).

% prod_cases4
tff(fact_37_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A4: A,B4: B,C2: 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),B4),C2)) ).

% prod_cases3
tff(fact_38_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: 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),A3),B3) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_39_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o),P2: product_prod(A,B)] :
      ( ! [A4: A,B4: B] : aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))
     => aa(product_prod(A,B),$o,P,P2) ) ).

% prod_cases
tff(fact_40_surj__pair,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B)] :
    ? [X4: A,Y3: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) ).

% surj_pair
tff(fact_41_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_42_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_43_add__self__div__2,axiom,
    ! [Mb: 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_44_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,$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: $tType,A5: set(A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_a(set(A),fun(A,$o),A5)) = A5 ).

% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
        <=> aa(A,$o,Q,X4) )
     => ( aa(fun(A,$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,F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X4: A] : aa(A,B,F3,X4) = aa(A,B,G3,X4)
     => ( F3 = G3 ) ) ).

% ext
tff(fact_48_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_49_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_50_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_51_nat__add__left__cancel__le,axiom,
    ! [K2: 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),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_52_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xb,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),divide_divide(A,Xb,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = Xb ) ).

% field_sum_of_halves
tff(fact_53_semiring__norm_I85_J,axiom,
    ! [Mb: num] : aa(num,num,bit0,Mb) != one2 ).

% semiring_norm(85)
tff(fact_54_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one2 != aa(num,num,bit0,Nb) ).

% semiring_norm(83)
tff(fact_55_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_56_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_57_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_58_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),S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,Nb)),TreeList,S) ) ).

% deg_SUcn_Node
tff(fact_59_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_60_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_61_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Ta),Xb)
      <=> member(nat,Xb,vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_62_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_63_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_64_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_65_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_66_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_67_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_68_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,$o),ord_less(nat),Xb),Y)
        & ! [Z2: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Ta),Z2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Z2) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Z2) ) ) ) ).

% succ_member
tff(fact_69_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_70_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_71_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_72_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_73_lessI,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(nat,nat,suc,Nb)) ).

% lessI
tff(fact_74_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_75_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_76_nat__add__left__cancel__less,axiom,
    ! [K2: 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),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_77_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_78_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_79_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_80_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_81_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_82_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_83_div2__Suc__Suc,axiom,
    ! [Mb: 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,divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% div2_Suc_Suc
tff(fact_84_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_85__C5_Ohyps_C_I10_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)) ).

% "5.hyps"(10)
tff(fact_86_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_87_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_88_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ~ ! [Q2: nat] : Nb != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q2)) ) ).

% less_natE
tff(fact_89_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,I)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),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_90_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),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,I) ) ) ) ).

% inc_induct
tff(fact_91_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_92_less__add__Suc1,axiom,
    ! [I: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Mb))) ).

% less_add_Suc1
tff(fact_93_less__add__Suc2,axiom,
    ! [I: nat,Mb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I))) ).

% less_add_Suc2
tff(fact_94_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_95_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_96_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_97_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_98_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_99_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ? [K: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K)) ) ).

% less_imp_Suc_add
tff(fact_100_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V2: fun(A,nat),Xb: A] :
      ( ! [X4: A] :
          ( ~ aa(A,$o,P,X4)
         => ? [Y4: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y4)),aa(A,nat,V2,X4))
              & ~ aa(A,$o,P,Y4) ) )
     => aa(A,$o,P,Xb) ) ).

% infinite_descent_measure
tff(fact_101_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_102_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X4: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Y4)),aa(A,B,F3,X4))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct_rule
tff(fact_103_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,P,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I2))
               => aa(nat,$o,P,I2) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_104_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_105_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_106_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_107_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_irrefl_nat
tff(fact_108_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),K) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_109_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X4: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Y4)),aa(A,B,F3,X4))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct
tff(fact_110_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K2) ) ) ).

% less_trans_Suc
tff(fact_111_less__not__refl3,axiom,
    ! [S2: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S2),Ta)
     => ( S2 != Ta ) ) ).

% less_not_refl3
tff(fact_112_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_113_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Nb) ).

% less_not_refl
tff(fact_114_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_115_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_116_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_117_All__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,Nb)
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
           => aa(nat,$o,P,I3) ) ) ) ).

% All_less_Suc
tff(fact_118_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_119_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_120_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa(nat,nat,suc,Nb) ).

% n_not_Suc_n
tff(fact_121_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_122_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,Nb)
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
            & aa(nat,$o,P,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_123_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xb) = aa(nat,nat,suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_124_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_125_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_126_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_127_Suc__lessE,axiom,
    ! [I: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K2)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
           => ( K2 != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_128_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_129_Nat_OlessE,axiom,
    ! [I: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K2)
     => ( ( K2 != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
             => ( K2 != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_130_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Nb: nat,Mb: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,aa(nat,nat,suc,N)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,Mb))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_131_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Nb: nat,N2: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,N2)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_132_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_133_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( ! [X4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,X4),X4)
       => ( ! [X4: nat,Y3: nat,Z3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X4),Y3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y3),Z3)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X4),Z3) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_nat__arith_Osuc1,axiom,
    ! [A5: nat,K2: nat,A2: nat] :
      ( ( A5 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),A2) )
     => ( aa(nat,nat,suc,A5) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_146_less__mono__imp__le__mono,axiom,
    ! [F3: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F3,I2)),aa(nat,nat,F3,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F3,I)),aa(nat,nat,F3,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_147_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_148_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_149_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_150_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_151_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_152_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),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),K2),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_153_trans__less__add2,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),J)) ) ).

% trans_less_add2
tff(fact_154_trans__less__add1,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Mb)) ) ).

% trans_less_add1
tff(fact_155_add__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) ) ).

% add_less_mono1
tff(fact_156_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I) ).

% not_add_less2
tff(fact_157_not__add__less1,axiom,
    ! [I: 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),I),J)),I) ).

% not_add_less1
tff(fact_158_add__less__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_159_add__lessD1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K2) ) ).

% add_lessD1
tff(fact_160_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_161_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Nb: nat,N2: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N))),aa(nat,A,F3,N))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,F3,Nb)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_162_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Nb: nat,N2: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,F3,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,N2)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_163_mono__nat__linear__lb,axiom,
    ! [F3: fun(nat,nat),Mb: nat,K2: 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,F3,M)),aa(nat,nat,F3,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,F3,Mb)),K2)),aa(nat,nat,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2))) ) ).

% mono_nat_linear_lb
tff(fact_164_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Mb,Nb)),divide_divide(nat,aa(nat,nat,suc,Mb),Nb)) ).

% Suc_div_le_mono
tff(fact_165_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,Xb,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Y)
        & ! [X3: nat] :
            ( member(nat,X3,Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),X3)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X3) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_166_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% is_num_normalize(1)
tff(fact_167_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K2: nat,B2: nat] :
      ( aa(nat,$o,P,K2)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => ? [X4: nat] :
            ( aa(nat,$o,P,X4)
            & ! [Y4: nat] :
                ( aa(nat,$o,P,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),X4) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_168_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_169_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_170_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_171_le__trans,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K2) ) ) ).

% le_trans
tff(fact_172_le__refl,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Nb) ).

% le_refl
tff(fact_173_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),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_174_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_175_trans__le__add2,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),J)) ) ).

% trans_le_add2
tff(fact_176_trans__le__add1,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),Mb)) ) ).

% trans_le_add1
tff(fact_177_add__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) ) ).

% add_le_mono1
tff(fact_178_add__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_179_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),L)
     => ? [N: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),N) ) ).

% le_Suc_ex
tff(fact_180_add__leD2,axiom,
    ! [Mb: nat,K2: 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),K2)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb) ) ).

% add_leD2
tff(fact_181_add__leD1,axiom,
    ! [Mb: nat,K2: 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),K2)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% add_leD1
tff(fact_182_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_183_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_184_add__leE,axiom,
    ! [Mb: nat,K2: 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),K2)),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),K2),Nb) ) ) ).

% add_leE
tff(fact_185_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Mb,Nb)),Mb) ).

% div_le_dividend
tff(fact_186_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K2: 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),divide_divide(nat,Mb,K2)),divide_divide(nat,Nb,K2)) ) ).

% div_le_mono
tff(fact_187_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_188_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] : divide_divide(A,A2,aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% divide_numeral_1
tff(fact_189_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : 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_190_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_191_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_192_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_193_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_194_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_195_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_196_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_right
tff(fact_197_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_left
tff(fact_198_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_199_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_200_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_201__C5_Ohyps_C_I7_J,axiom,
    ! [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))),m))
     => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I4)),X_12)
      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(summary),I4) ) ) ).

% "5.hyps"(7)
tff(fact_202_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_203_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C3: 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),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add_left_cancel
tff(fact_204_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C3: 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),C3),A2) )
        <=> ( B2 = C3 ) ) ) ).

% add_right_cancel
tff(fact_205_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_206_pow__sum,axiom,
    ! [A2: nat,B2: 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_207_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_208_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_209_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_210_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_211_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_212_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_213__C5_Ohyps_C_I4_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) ).

% "5.hyps"(4)
tff(fact_214_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_215_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [A: $tType,F3: 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,F3),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),F3,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_216_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_217_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K2: A,L: A] :
          ( ( ( I = J )
            & ( K2 = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_218_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: A,K2: A,A2: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_219_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B5: A,K2: A,B2: A,A2: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_220_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% add.assoc
tff(fact_221_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: 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),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add.left_cancel
tff(fact_222_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A2: A,C3: 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),C3),A2) )
        <=> ( B2 = C3 ) ) ) ).

% add.right_cancel
tff(fact_223_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_224_add_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% add.left_commute
tff(fact_225_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C3: 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),C3) )
         => ( B2 = C3 ) ) ) ).

% add_left_imp_eq
tff(fact_226_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C3: 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),C3),A2) )
         => ( B2 = C3 ) ) ) ).

% add_right_imp_eq
tff(fact_227_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & ( K2 = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_228_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K2: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_229_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_230_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_mono
tff(fact_231_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% add_left_mono
tff(fact_232_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)
         => ~ ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ) ).

% less_eqE
tff(fact_233_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ) ).

% add_right_mono
tff(fact_234_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_235_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_236_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_237_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_238_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K2: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_239_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & ( K2 = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_240_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_strict_mono
tff(fact_241_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% add_strict_left_mono
tff(fact_242_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ) ).

% add_strict_right_mono
tff(fact_243_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_left
tff(fact_244_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_right
tff(fact_245_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_246_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K2: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K2),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_247_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_le_less_mono
tff(fact_248_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) ) ) ) ).

% add_less_le_mono
tff(fact_249_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),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% mintlistlength
tff(fact_250__092_060open_062length_AtreeList_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_Am_092_060close_062,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) )
    & vEBT_invar_vebt(summary,m) ) ).

% \<open>length treeList = 2 ^ m \<and> invar_vebt summary m\<close>
tff(fact_251_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Mb: nat,Nb: nat] : divide_divide(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)) = 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_252__C5_Ohyps_C_I11_J,axiom,
    ( ( mi != ma )
   => ! [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))),m))
       => ( ( ( vEBT_VEBT_high(ma,na) = I4 )
           => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I4)),vEBT_VEBT_low(ma,na)) )
          & ! [X: nat] :
              ( ( ( vEBT_VEBT_high(X,na) = I4 )
                & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I4)),vEBT_VEBT_low(X,na)) )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),mi),X)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),ma) ) ) ) ) ) ).

% "5.hyps"(11)
tff(fact_253_self__le__ge2__pow,axiom,
    ! [K2: 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))),K2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K2),Mb)) ) ).

% self_le_ge2_pow
tff(fact_254_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_255_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_256_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_257_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_258_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_259_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)) = 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_260_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Nb: nat] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) )
     => ( aa(nat,$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_261_bit__split__inv,axiom,
    ! [Xb: nat,D3: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xb,D3),vEBT_VEBT_low(Xb,D3),D3) = Xb ).

% bit_split_inv
tff(fact_262_high__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_high(Xb,Nb) = 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_263_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_264_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_265_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C3: 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),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_right
tff(fact_266_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_267_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C3: 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),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_left
tff(fact_268_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_269_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_270_Suc__diff__diff,axiom,
    ! [Mb: nat,Nb: nat,K2: 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,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),K2) ).

% Suc_diff_diff
tff(fact_271_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_272_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),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),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_273_diff__diff__left,axiom,
    ! [I: nat,J: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) ).

% diff_diff_left
tff(fact_274_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_275_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K2) ) ) ).

% Nat.add_diff_assoc2
tff(fact_276_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K2) ) ) ).

% Nat.add_diff_assoc
tff(fact_277_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,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_278_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,$o),ord_less(nat),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ) ).

% member_inv
tff(fact_279_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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),K2))),I) = 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),K2),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_280_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_281_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),Nb: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M2),N)
             => aa(extended_enat,$o,P,M2) )
         => aa(extended_enat,$o,P,N) )
     => aa(extended_enat,$o,P,Nb) ) ).

% enat_less_induct
tff(fact_282_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A,D3: 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),C3),D3) )
         => ( ( A2 = B2 )
          <=> ( C3 = D3 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_283_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C3: 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),C3)),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)),C3) ) ).

% diff_right_commute
tff(fact_284_diff__commute,axiom,
    ! [I: nat,J: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),K2)),J) ).

% diff_commute
tff(fact_285_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_286_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_287_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) ) ) ).

% diff_right_mono
tff(fact_288_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$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),C3),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ) ).

% diff_left_mono
tff(fact_289_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ) ) ).

% diff_mono
tff(fact_290_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) ) ) ).

% diff_strict_right_mono
tff(fact_291_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$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),C3),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_292_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3) ) ) ) ).

% diff_eq_diff_less
tff(fact_293_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ) ) ).

% diff_strict_mono
tff(fact_294_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C3: A,B2: A,D3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3)) = 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),C3),D3)) ) ).

% add_diff_add
tff(fact_295_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% diff_diff_eq
tff(fact_296_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) = A2 )
         => ( C3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_297_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_298_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ).

% diff_add_eq
tff(fact_299_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ).

% diff_diff_eq2
tff(fact_300_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) ) ).

% add_diff_eq
tff(fact_301_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = C3 ) ) ) ).

% eq_diff_eq
tff(fact_302_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = C3 )
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) ) ) ) ).

% diff_eq_eq
tff(fact_303_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A5: A,K2: A,A2: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A5),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_304_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K2: nat,I: nat] :
      ( aa(nat,$o,P,K2)
     => ( ! [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),K2),I)) ) ) ).

% zero_induct_lemma
tff(fact_305_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Nb)),K2) ) ).

% less_imp_diff_less
tff(fact_306_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_307_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_308_le__diff__iff_H,axiom,
    ! [A2: nat,C3: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),C3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),A2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_309_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_310_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_311_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_312_le__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_313_eq__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_314_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_315_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_316_diff__cancel2,axiom,
    ! [Mb: nat,K2: 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),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% diff_cancel2
tff(fact_317_Nat_Odiff__cancel,axiom,
    ! [K2: 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),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb) ).

% Nat.diff_cancel
tff(fact_318_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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) = C3 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_319_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_320_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),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),C3),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_321_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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),C3)),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)),C3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_322_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_323_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$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),C3),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_324_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),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),C3),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_325_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_326_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A2)) ) ) ).

% le_add_diff
tff(fact_327_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_328_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) ) ) ).

% le_diff_eq
tff(fact_329_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% diff_le_eq
tff(fact_330_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% diff_less_eq
tff(fact_331_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) ) ) ).

% less_diff_eq
tff(fact_332_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_333_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_334_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_335_less__diff__iff,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Mb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_336_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C3: 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),C3),A2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),C3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C3)) ) ) ).

% diff_less_mono
tff(fact_337_less__diff__conv,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)),J) ) ).

% less_diff_conv
tff(fact_338_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_339_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) = K2 )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_340_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_341_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)) ) ) ).

% Nat.diff_add_assoc
tff(fact_342_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_343_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I: 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),K2)),I)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)) ) ).

% le_diff_conv
tff(fact_344_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_345_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K2)),I)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2)) ) ) ).

% less_diff_conv2
tff(fact_346_diff__le__diff__pow,axiom,
    ! [K2: 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))),K2)
     => 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),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K2),Nb))) ) ).

% diff_le_diff_pow
tff(fact_347_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X4,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) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X4: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( vEBT_VEBT_high(X4,Nb) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(X4,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),X4)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),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_348_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: nat,Maa: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X4,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) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I2) ) )
               => ( ( ( Mia = Maa )
                   => ! [X4: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Mb))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X4: nat] :
                                    ( ( ( vEBT_VEBT_high(X4,Nb) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I2)),vEBT_VEBT_low(X4,Nb)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mia),X4)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),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_349_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,A2,B2)),Nb) = 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_350_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_351_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,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,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => aa(nat,$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_352_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,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(Xb,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_353_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,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% summaxma
tff(fact_354_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,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_355_set__n__deg__not__0,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Mb: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X4,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_356_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_357_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_358_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_359_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_360_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))
     => member(A,aa(nat,A,nth(A,Xs),Nb),aa(list(A),set(A),set2(A),Xs)) ) ).

% nth_mem
tff(fact_361__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),na) ).

% \<open>1 \<le> n\<close>
tff(fact_362_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_363_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_364_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_365_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_366_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,A2,one_one(A)) = A2 ) ).

% div_by_1
tff(fact_367_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : divide_divide(A,A2,one_one(A)) = A2 ) ).

% bits_div_by_1
tff(fact_368_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_369_bit__concat__def,axiom,
    ! [H: nat,L: nat,D3: nat] : vEBT_VEBT_bit_concat(H,L,D3) = 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))),D3))),L) ).

% bit_concat_def
tff(fact_370_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_371_not__Some__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ! [Y5: A] : Xb != aa(A,option(A),some(A),Y5)
    <=> ( Xb = none(A) ) ) ).

% not_Some_eq
tff(fact_372_not__None__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ( Xb != none(A) )
    <=> ? [Y5: A] : Xb = aa(A,option(A),some(A),Y5) ) ).

% not_None_eq
tff(fact_373_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_374_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_375_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: A,C3: 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),C3)) = 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)),C3)) ) ).

% distrib_left_numeral
tff(fact_376_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_377_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,B2: A,C3: 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),C3)) = 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)),C3)) ) ).

% right_diff_distrib_numeral
tff(fact_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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,$o),ord_less_eq(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$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_386_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,$o),ord_less_eq(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$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_387_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,$o),ord_less(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$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_388_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,$o),ord_less(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$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_389_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_390_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_391_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_392_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_393_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_394_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_395_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))),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,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_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xb: A] :
          ( ( one_one(A) = Xb )
        <=> ( Xb = one_one(A) ) ) ) ).

% one_reorient
tff(fact_405_mult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A2: A,C3: 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),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% mult.left_commute
tff(fact_406_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_407_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_408_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% mult.assoc
tff(fact_409_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_410_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_411_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_412_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_413_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% ring_class.ring_distribs(2)
tff(fact_414_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = 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),C3)) ) ).

% ring_class.ring_distribs(1)
tff(fact_415_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% comm_semiring_class.distrib
tff(fact_416_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = 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),C3)) ) ).

% distrib_left
tff(fact_417_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% distrib_right
tff(fact_418_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,E3: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),C3)) = 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)),E3)),C3) ) ).

% combine_common_factor
tff(fact_419_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% left_diff_distrib
tff(fact_420_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = 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),C3)) ) ).

% right_diff_distrib
tff(fact_421_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C3: 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),C3)),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),C3),A2)) ) ).

% left_diff_distrib'
tff(fact_422_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = 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),C3)) ) ).

% right_diff_distrib'
tff(fact_423_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_424_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_425_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_426_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_427_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_428_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_429_Suc__mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_430_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))] :
      ( ! [Uu: fun(A,fun(A,A)),Uv: 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))),Uu),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)),Uv))
     => ( ! [Uw: fun(A,fun(A,A)),V3: 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))),Uw),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F2: fun(A,fun(A,A)),A4: A,B4: 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))),F2),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),B4))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_431_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))] :
      ( ! [Uu: fun(A,fun(A,$o)),Uv: option(A)] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$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,$o)),product_prod(option(A),option(A))),Uu),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv))
     => ( ! [Uw: fun(A,fun(A,$o)),V3: 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))),Uw),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F2: fun(A,fun(A,$o)),X4: A,Y3: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$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))),F2),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),X4)),aa(A,option(A),some(A),Y3))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_432_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,B4: B] :
              ( ( Xb = aa(A,option(A),some(A),A4) )
             => ( ( Y = aa(B,option(B),some(B),B4) )
               => 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_433_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ! [X_12: option(A)] : aa(option(A),$o,P,X_12)
    <=> ( aa(option(A),$o,P,none(A))
        & ! [X3: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X3)) ) ) ).

% split_option_all
tff(fact_434_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ? [X_12: option(A)] : aa(option(A),$o,P,X_12)
    <=> ( aa(option(A),$o,P,none(A))
        | ? [X3: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X3)) ) ) ).

% split_option_ex
tff(fact_435_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_436_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X2: A] :
      ( ( Option = aa(A,option(A),some(A),X2) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_437_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X2: A] : none(A) != aa(A,option(A),some(A),X2) ).

% option.distinct(1)
tff(fact_438_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_439_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_440_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_441_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_442_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_443_mult__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_444_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K2)) ) ).

% mult_le_mono1
tff(fact_445_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),J)) ) ).

% mult_le_mono2
tff(fact_446_add__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: 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)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2)) ).

% add_mult_distrib
tff(fact_447_add__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),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),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)) ).

% add_mult_distrib2
tff(fact_448_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K2) ).

% left_add_mult_distrib
tff(fact_449_diff__mult__distrib2,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),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),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)) ).

% diff_mult_distrib2
tff(fact_450_diff__mult__distrib,axiom,
    ! [Mb: nat,Nb: nat,K2: 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)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2)) ).

% diff_mult_distrib
tff(fact_451_div__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : divide_divide(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q3)) = divide_divide(nat,divide_divide(nat,Mb,Nb),Q3) ).

% div_mult2_eq
tff(fact_452_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( 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)),E3)),C3) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_453_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3) )
        <=> ( C3 = 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)),E3)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_454_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_455_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_456_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_457_option_Osel,axiom,
    ! [A: $tType,X2: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X2)) = X2 ).

% option.sel
tff(fact_458_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_459_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,A)),Uv2: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uu2),none(A)),Uv2) = none(A) ).

% VEBT_internal.option_shift.simps(1)
tff(fact_460_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
        <=> aa(A,$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)),E3)),C3)),D3) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_461_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),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)),E3)),D3)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_462_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),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)),E3)),D3)) ) ) ).

% less_add_iff2
tff(fact_463_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D3))
        <=> aa(A,$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)),E3)),C3)),D3) ) ) ).

% less_add_iff1
tff(fact_464_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_465_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_466_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_467_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_468_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_469_Suc__mult__less__cancel1,axiom,
    ! [K2: 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,K2)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_470_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_471_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_472_Suc__mult__le__cancel1,axiom,
    ! [K2: 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,K2)),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_473_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_474_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_475_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_476_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_477_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_478_less__mult__imp__div__less,axiom,
    ! [Mb: nat,I: 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),I),Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Mb,Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_479_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,one_one(A),A2)),Nb) = divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_one_over
tff(fact_480_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),divide_divide(nat,Mb,Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_481_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),divide_divide(nat,Mb,Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_482_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_483_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_484_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_485_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_486_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_487_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_488_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) ) )
       => ( ( ? [V3: A] : Xaa = aa(A,option(A),some(A),V3)
           => ( ( Xba = none(A) )
             => ( Y != none(A) ) ) )
         => ~ ! [A4: A] :
                ( ( Xaa = aa(A,option(A),some(A),A4) )
               => ! [B4: A] :
                    ( ( Xba = aa(A,option(A),some(A),B4) )
                   => ( Y != aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Xb,A4),B4)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_489_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw2: fun(A,fun(A,A)),V: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uw2),aa(A,option(A),some(A),V)),none(A)) = none(A) ).

% VEBT_internal.option_shift.simps(2)
tff(fact_490_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_491_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_492_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_493_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
         => ( aa(A,$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),N3)) ) ) ) ).

% power_strict_increasing
tff(fact_494_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),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),I),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_495_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),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),I)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_496_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( 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),I),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),I),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_497_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),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),I),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),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_498_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( 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),I),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),I),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_499_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),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),I),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),I)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_500_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
           => aa(A,$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),N3)) ) ) ) ).

% power_increasing
tff(fact_501_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B5)
    <=> ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => member(A,X3,B5) ) ) ).

% subset_code(1)
tff(fact_502_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_503_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_504_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_505_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_506_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_507_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_508_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_509_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_510_div__nat__eqI,axiom,
    ! [Nb: nat,Q3: 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),Q3)),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,Q3)))
       => ( divide_divide(nat,Mb,Nb) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_511_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_512_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( 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),I),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),I),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_513_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),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),I),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),I)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_514_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_515_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_516_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_517_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_518_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_519_ex__power__ivl1,axiom,
    ! [B2: nat,K2: 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)),K2)
       => ? [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)),K2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),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_520_ex__power__ivl2,axiom,
    ! [B2: nat,K2: 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))),K2)
       => ? [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)),K2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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_521_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K2: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K2)) ) ) ).

% add_le_imp_le_diff
tff(fact_522_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K2: A,Nb: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),K2)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_523_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_524_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I3: nat] :
            ( aa(nat,$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) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_525_Skolem__list__nth,axiom,
    ! [A: $tType,K2: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K2)
         => ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),P,I3),X_12) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K2 )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K2)
             => aa(A,$o,aa(nat,fun(A,$o),P,I3),aa(nat,A,nth(A,Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_526_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_527_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) )
    <=> ! [I3: nat] :
          ( aa(nat,$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)) ) ) ).

% all_set_conv_all_nth
tff(fact_528_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xb: A] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_529_in__set__conv__nth,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [I3: nat] :
          ( aa(nat,$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) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_530_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))
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P,X4) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_531_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_532_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_533_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_534_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Ta)
     => ( vEBT_vebt_mint(Ta) = none(nat) ) ) ).

% minNullmin
tff(fact_535_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none(nat) )
     => vEBT_VEBT_minNull(Ta) ) ).

% minminNull
tff(fact_536_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_537_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_538_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C3: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C3)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C3) ) ).

% times_divide_eq_left
tff(fact_539_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,divide_divide(A,A2,B2),C3) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% divide_divide_eq_left
tff(fact_540_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,A2,divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),B2) ) ).

% divide_divide_eq_right
tff(fact_541_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ).

% times_divide_eq_right
tff(fact_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C3: real,B2: real,D3: 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),C3))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D3))),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),D3),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),C3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_551_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,K2: num,L: num] : divide_divide(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),K2)),aa(num,A,numeral_numeral(A),L)) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K2),L))) ) ).

% div_mult2_numeral_eq
tff(fact_552_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_553_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
        ? [X_13: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),X_13) ) ).

% linordered_field_no_ub
tff(fact_554_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X) ) ).

% linordered_field_no_lb
tff(fact_555_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_556_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) = none(nat) ).

% vebt_mint.simps(2)
tff(fact_557_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) = none(nat) ).

% vebt_maxt.simps(2)
tff(fact_558_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,divide_divide(A,A2,B2),C3) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).

% divide_divide_eq_left'
tff(fact_559_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Z: A,W: A] : divide_divide(A,divide_divide(A,Xb,Y),divide_divide(A,Z,W)) = 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_560_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),divide_divide(A,Xb,Y)),divide_divide(A,Z,W)) = 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_561_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ).

% add_divide_distrib
tff(fact_562_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ).

% diff_divide_distrib
tff(fact_563_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X4,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,$o,vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X4: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_564_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X4,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,$o,vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X4: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X4,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X4),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Dega,TreeLista,Summarya),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_565_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),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_566_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),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_567_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_568_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa(real,real,aa(real,fun(real,real),minus_minus(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) = 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_569_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa(real,real,aa(real,fun(real,real),minus_minus(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) = 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_570_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,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,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,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))),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))),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_571_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,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xb,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))),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))),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_572_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q3: A,R2: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q3)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),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))),Q3)),R2)) ) ).

% divmod_step_eq
tff(fact_573_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_574_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_575_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_576_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_577_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_578_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : 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))) = divide_divide(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_579_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_580_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_581_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_582_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_583_real__divide__square__eq,axiom,
    ! [R2: real,A2: real] : divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),R2),A2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),R2)) = divide_divide(real,A2,R2) ).

% real_divide_square_eq
tff(fact_584_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_585_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: 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),C3)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self4
tff(fact_586_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: 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),C3),B2)),A2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self3
tff(fact_587_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C3: 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),C3)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self2
tff(fact_588_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: 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),C3),B2)),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mult_self1
tff(fact_589_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K2: 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),K2)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self4
tff(fact_590_Suc__mod__mult__self3,axiom,
    ! [K2: 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),K2),Nb)),Mb)),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self3
tff(fact_591_Suc__mod__mult__self2,axiom,
    ! [Mb: nat,Nb: nat,K2: 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),K2))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self2
tff(fact_592_Suc__mod__mult__self1,axiom,
    ! [Mb: nat,K2: 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),K2),Nb))),Nb) = modulo_modulo(nat,aa(nat,nat,suc,Mb),Nb) ).

% Suc_mod_mult_self1
tff(fact_593_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_594_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_595_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_596_Suc__times__numeral__mod__eq,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K2) != 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),K2)),Nb)),aa(num,nat,numeral_numeral(nat),K2)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_597_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_598_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C3: A] : aTP_Lamp_aa(A,fun(A,A),C3) = aa(A,fun(A,A),times_times(A),C3) ) ).

% mult_commute_abs
tff(fact_599_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ab(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_600_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ).

% mod_mult_right_eq
tff(fact_601_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C3)),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ).

% mod_mult_left_eq
tff(fact_602_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C3),modulo_modulo(A,A2,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ).

% mult_mod_right
tff(fact_603_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,B2)),C3) ) ).

% mod_mult_mult2
tff(fact_604_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,A3: A,B2: A,B3: A] :
          ( ( modulo_modulo(A,A2,C3) = modulo_modulo(A,A3,C3) )
         => ( ( modulo_modulo(A,B2,C3) = modulo_modulo(A,B3,C3) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),C3) ) ) ) ) ).

% mod_mult_cong
tff(fact_605_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C3)),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ).

% mod_mult_eq
tff(fact_606_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C3)),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) ) ).

% mod_add_eq
tff(fact_607_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,A3: A,B2: A,B3: A] :
          ( ( modulo_modulo(A,A2,C3) = modulo_modulo(A,A3,C3) )
         => ( ( modulo_modulo(A,B2,C3) = modulo_modulo(A,B3,C3) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),C3) ) ) ) ) ).

% mod_add_cong
tff(fact_608_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C3)),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) ) ).

% mod_add_left_eq
tff(fact_609_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) ) ).

% mod_add_right_eq
tff(fact_610_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C3) ) ).

% mod_diff_right_eq
tff(fact_611_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,C3)),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C3) ) ).

% mod_diff_left_eq
tff(fact_612_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C3: A,A3: A,B2: A,B3: A] :
          ( ( modulo_modulo(A,A2,C3) = modulo_modulo(A,A3,C3) )
         => ( ( modulo_modulo(A,B2,C3) = modulo_modulo(A,B3,C3) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B3),C3) ) ) ) ) ).

% mod_diff_cong
tff(fact_613_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A2,C3)),modulo_modulo(A,B2,C3)),C3) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C3) ) ).

% mod_diff_eq
tff(fact_614_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_615_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_616_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_617_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_618_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q3: 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,Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q3)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q3)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_619_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_620_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_621_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_622_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_623_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_624_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_625_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q3: 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,Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) ) ).

% cong_exp_iff_simps(8)
tff(fact_626_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: num,Nb: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) ) ).

% cong_exp_iff_simps(6)
tff(fact_627_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( modulo_modulo(A,A2,C3) = modulo_modulo(A,B2,C3) )
         => ~ ! [D2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D2)) ) ) ).

% mod_eqE
tff(fact_628_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3))),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,C3)),modulo_modulo(A,B2,C3)),C3)) ) ).

% div_add1_eq
tff(fact_629_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_630_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_631_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Mb,Q3) = modulo_modulo(nat,Nb,Q3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
       => ~ ! [S: 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),Q3),S)) ) ) ).

% mod_eq_nat1E
tff(fact_632_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( ( modulo_modulo(nat,Mb,Q3) = modulo_modulo(nat,Nb,Q3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
       => ~ ! [S: 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),Q3),S)) ) ) ).

% mod_eq_nat2E
tff(fact_633_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)
       => ? [Q2: 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),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_634_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_635_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_636_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_ac(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_637_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A,C3: 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),divide_divide(A,A2,B2))),modulo_modulo(A,A2,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) ) ).

% cancel_div_mod_rules(2)
tff(fact_638_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A,C3: 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),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2))),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) ) ).

% cancel_div_mod_rules(1)
tff(fact_639_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),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2)) ) ).

% mod_div_decomp
tff(fact_640_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),divide_divide(A,A2,B2)),B2)),modulo_modulo(A,A2,B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_641_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),divide_divide(A,A2,B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_642_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),divide_divide(A,A2,B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_643_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),divide_divide(A,A2,B2))),modulo_modulo(A,A2,B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_644_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: A,C3: A] : divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C3))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C3)),C3)) ) ).

% div_mult1_eq
tff(fact_645_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),divide_divide(A,A2,B2)),B2)) = modulo_modulo(A,A2,B2) ) ).

% minus_div_mult_eq_mod
tff(fact_646_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),divide_divide(A,A2,B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_647_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),divide_divide(A,A2,B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_648_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),divide_divide(A,A2,B2))) = modulo_modulo(A,A2,B2) ) ).

% minus_mult_div_eq_mod
tff(fact_649_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : modulo_modulo(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q3)) = 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,divide_divide(nat,Mb,Nb),Q3))),modulo_modulo(nat,Mb,Nb)) ).

% mod_mult2_eq
tff(fact_650_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),divide_divide(nat,Mb,Nb)),Nb)) ).

% modulo_nat_def
tff(fact_651_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat,Mb: nat] : modulo_modulo(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)) = 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_652_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_653_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,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,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))),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))),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_654_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,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,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_655_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_656_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% Diff_eq_empty_iff
tff(fact_657_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,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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_658_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeLista: list(vEBT_VEBT),S2: vEBT_VEBT,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeLista,S2),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_659_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,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,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),pos),vEBT_VEBT_low(Xb,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_660_div__mod__decomp,axiom,
    ! [A5: nat,Nb: nat] : A5 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,A5,Nb)),Nb)),modulo_modulo(nat,A5,Nb)) ).

% div_mod_decomp
tff(fact_661_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),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% set_bit_Suc
tff(fact_662_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,divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% flip_bit_Suc
tff(fact_663_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),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% unset_bit_Suc
tff(fact_664_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_665_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_leaf
tff(fact_666_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_667_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( aa(num,num,bit0,X2) = aa(num,num,bit0,Y2) )
    <=> ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_668_Diff__idemp,axiom,
    ! [A: $tType,A5: set(A),B5: 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)),A5),B5)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) ).

% Diff_idemp
tff(fact_669_Diff__iff,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
    <=> ( member(A,C3,A5)
        & ~ member(A,C3,B5) ) ) ).

% Diff_iff
tff(fact_670_DiffI,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,A5)
     => ( ~ member(A,C3,B5)
       => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) ) ) ).

% DiffI
tff(fact_671_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_672_Diff__empty,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),bot_bot(set(A))) = A5 ).

% Diff_empty
tff(fact_673_empty__Diff,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A5) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_674_Diff__cancel,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),A5) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_675_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_676_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ? [B4: A] : member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) ) ).

% psubset_imp_ex_mem
tff(fact_677_minus__set__def,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = aa(fun(A,$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)),aTP_Lamp_a(set(A),fun(A,$o),A5)),aTP_Lamp_a(set(A),fun(A,$o),B5))) ).

% minus_set_def
tff(fact_678_set__diff__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(set(A),fun(A,$o)),A5),B5)) ).

% set_diff_eq
tff(fact_679_DiffD2,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
     => ~ member(A,C3,B5) ) ).

% DiffD2
tff(fact_680_DiffD1,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
     => member(A,C3,A5) ) ).

% DiffD1
tff(fact_681_DiffE,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
     => ~ ( member(A,C3,A5)
         => member(A,C3,B5) ) ) ).

% DiffE
tff(fact_682_div__mod__decomp__int,axiom,
    ! [A5: int,Nb: int] : A5 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),divide_divide(int,A5,Nb)),Nb)),modulo_modulo(int,A5,Nb)) ).

% div_mod_decomp_int
tff(fact_683_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_684_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_685_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_686_double__diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A5)) = A5 ) ) ) ).

% double_diff
tff(fact_687_Diff__subset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),$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)),A5),B5)),A5) ).

% Diff_subset
tff(fact_688_Diff__mono,axiom,
    ! [A: $tType,A5: set(A),C5: set(A),D4: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D4),B5)
       => aa(set(A),$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)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),D4)) ) ) ).

% Diff_mono
tff(fact_689_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% verit_comp_simplify1(3)
tff(fact_690_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one2 != aa(num,num,bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_691_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xb) ).

% vebt_member.simps(2)
tff(fact_692_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_693_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_694_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,V3: 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,V3),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) )
         => ~ ! [V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_695_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( ! [Uu: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((Uu),(B4)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != $ite((B4),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
           => ( ? [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) ) )
           => ( ( ? [V3: 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)),V3),zero_zero(nat),Vc2,Vd2)
               => ( Y != none(nat) ) )
             => ( ( ? [V3: 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)),V3),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,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,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))),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))),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_696_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),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_697_triangle__def,axiom,
    ! [Nb: nat] : nat_triangle(Nb) = 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_698_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A5: set(nat),B5: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Z)
     => ( vEBT_VEBT_max_in_set(A5,Z)
       => ( aa(set(nat),$o,finite_finite(nat),B5)
         => ( ( A5 = B5 )
           => ? [X_13: nat] : vEBT_is_succ_in_set(A5,Xb,X_13) ) ) ) ) ).

% obtain_set_succ
tff(fact_699_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_700_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_701_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,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,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,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xb,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,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Xb,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_702_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,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,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,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Mia,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,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Mia,divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_703_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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% set_bit_0
tff(fact_704_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_0_not
tff(fact_705_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_706_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_707_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_708_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
    <=> ? [A6: $o,B6: $o] : Ta = vEBT_Leaf((A6),(B6)) ) ).

% deg1Leaf
tff(fact_709_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
     => ? [A4: $o,B4: $o] : Ta = vEBT_Leaf((A4),(B4)) ) ).

% deg_1_Leaf
tff(fact_710_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( Nb = one_one(nat) )
       => ? [A4: $o,B4: $o] : Ta = vEBT_Leaf((A4),(B4)) ) ) ).

% deg_1_Leafy
tff(fact_711_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(set(nat),$o,finite_finite(nat),vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_712_succ__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_13: nat] : vEBT_is_succ_in_set(Xs,A2,X_13)
     => ( aa(set(nat),$o,finite_finite(nat),Xs)
       => ~ ? [X: nat] :
              ( member(nat,X,Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),X) ) ) ) ).

% succ_none_empty
tff(fact_713_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_714_list__update__overwrite,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xb: A,Y: A] : list_update(A,list_update(A,Xs,I,Xb),I,Y) = list_update(A,Xs,I,Y) ).

% list_update_overwrite
tff(fact_715_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_716_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_717_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_718_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_719_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_720_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_721_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_722_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_723_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_724_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_725_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_726_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_727_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_728_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_729_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_730_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_731_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_732_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_733_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_734_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_735_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_736_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ).

% div_0
tff(fact_737_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_738_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_739_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_740_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( divide_divide(A,A2,B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_741_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( divide_divide(A,C3,A2) = divide_divide(A,C3,B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_742_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( divide_divide(A,A2,C3) = divide_divide(A,B2,C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_743_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : divide_divide(A,A2,zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_744_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_745_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_746_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_747_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_748_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_749_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_750_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_751_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_752_le0,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% le0
tff(fact_753_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),$o,finite_finite(A),aa(list(A),set(A),set2(A),Xs)) ).

% List.finite_set
tff(fact_754_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_755_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_756_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_757_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_758_mult__cancel2,axiom,
    ! [Mb: nat,K2: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),K2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_759_mult__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb) )
    <=> ( ( Mb = Nb )
        | ( K2 = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_760_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_761_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_762_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xb: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,I,Xb)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_list__update__id,axiom,
    ! [A: $tType,Xs: list(A),I: nat] : list_update(A,Xs,I,aa(nat,A,nth(A,Xs),I)) = Xs ).

% list_update_id
tff(fact_771_nth__list__update__neq,axiom,
    ! [A: $tType,I: nat,J: nat,Xs: list(A),Xb: A] :
      ( ( I != J )
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xb)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_772_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B2: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_788_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_789_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B2: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_790_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_791_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_792_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_793_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_794_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(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_795_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(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_796_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = $ite(C3 = zero_zero(A),zero_zero(A),divide_divide(A,A2,B2)) ) ).

% div_mult_mult1_if
tff(fact_797_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult2
tff(fact_798_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% div_mult_mult1
tff(fact_799_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = $ite(C3 = zero_zero(A),zero_zero(A),divide_divide(A,A2,B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_800_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_801_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_802_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_803_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_804_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_805_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( divide_divide(A,A2,B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_806_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = divide_divide(A,A2,B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_807_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_808_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          divide_divide(A,A2,A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_809_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( divide_divide(A,B2,A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_810_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = divide_divide(A,B2,A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_811_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( divide_divide(A,one_one(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_812_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = divide_divide(A,one_one(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_813_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_814_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K2: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),K2)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_815_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_816_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_817_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_818_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_819_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : divide_divide(A,modulo_modulo(A,A2,B2),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_820_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : divide_divide(A,modulo_modulo(A,A2,B2),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_821_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_822_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_823_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_824_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_825_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_826_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_827_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_828_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_829_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_830_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_831_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_832_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_833_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_834_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_835_mult__less__cancel2,axiom,
    ! [Mb: nat,K2: 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),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_836_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_837_nat__mult__less__cancel__disj,axiom,
    ! [K2: 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),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_838_div__by__Suc__0,axiom,
    ! [Mb: nat] : divide_divide(nat,Mb,aa(nat,nat,suc,zero_zero(nat))) = Mb ).

% div_by_Suc_0
tff(fact_839_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_840_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_841_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_842_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
     => ( divide_divide(nat,Mb,Nb) = zero_zero(nat) ) ) ).

% div_less
tff(fact_843_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_844_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_845_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_846_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_847_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)) = $ite(K2 = zero_zero(nat),zero_zero(nat),divide_divide(nat,Mb,Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_848_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( list_update(A,Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_849_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_850_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_851_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_852_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_853_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_854_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,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_855_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,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_856_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,$o),ord_less(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_857_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,$o),ord_less(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_858_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_859_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: 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_860_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = 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_861_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_862_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,B2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_863_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( B2 != zero_zero(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),C3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self1
tff(fact_864_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( B2 != zero_zero(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),C3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self2
tff(fact_865_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( B2 != zero_zero(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),C3),B2)),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self3
tff(fact_866_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( B2 != zero_zero(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),C3)),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A2,B2)) ) ) ) ).

% div_mult_self4
tff(fact_867_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_868_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_869_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_870_nat__mult__le__cancel__disj,axiom,
    ! [K2: 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),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_871_mult__le__cancel2,axiom,
    ! [Mb: nat,K2: 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),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_872_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)
     => ( 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_873_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)
     => ( 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_874_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_875_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,$o),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_876_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,$o),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_877_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,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_878_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,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_879_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_880_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_881_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_882_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_883_set__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$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,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_884_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(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_885_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(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_886_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_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_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_895_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_896_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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% unset_bit_0
tff(fact_897_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: A] :
          ( ( zero_zero(A) = Xb )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_898_list__update__swap,axiom,
    ! [A: $tType,I: nat,I5: nat,Xs: list(A),Xb: A,X5: A] :
      ( ( I != I5 )
     => ( list_update(A,list_update(A,Xs,I,Xb),I5,X5) = list_update(A,list_update(A,Xs,I5,X5),I,Xb) ) ) ).

% list_update_swap
tff(fact_899_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_900_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_901_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_902_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X4: 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),(B4))),X4)
     => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: 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(Uu,zero_zero(nat),Uv,Uw)),Ux2)
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT,X4: 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,V3),TreeList,S)),X4) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_903_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_904_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_905_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_906_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_907_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_908_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_909_nat__add__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: 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)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Q3)) ).

% nat_add_max_left
tff(fact_910_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: 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),Q3)) = 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),Q3)) ).

% nat_add_max_right
tff(fact_911_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_912_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: 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),Q3)) = 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),Q3)) ).

% nat_mult_max_right
tff(fact_913_nat__mult__max__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: 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)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q3)) ).

% nat_mult_max_left
tff(fact_914_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu: $o,Uv: $o,D2: 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((Uu),(Uv))),D2)
     => ~ ! [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_915_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_916_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_917_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_918_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_919_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_920_finite__list,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A5 ) ).

% finite_list
tff(fact_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_934_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_935_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_936_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_937_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_938_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_939_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_940_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_941_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_942_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_943_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero(nat) != aa(nat,nat,suc,X2) ).

% nat.distinct(1)
tff(fact_944_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_945_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_946_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_947_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_948_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_949_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),Mb: nat,Nb: nat] :
      ( ! [X4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X4),zero_zero(nat))
     => ( ! [Y3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y3))
       => ( ! [X4: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X4),Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X4)),aa(nat,nat,suc,Y3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_950_zero__induct,axiom,
    ! [P: fun(nat,$o),K2: nat] :
      ( aa(nat,$o,P,K2)
     => ( ! [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_951_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa(nat,nat,suc,Mb) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_952_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_neq_Suc
tff(fact_953_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero(nat) != aa(nat,nat,suc,Mb) ).

% Zero_not_Suc
tff(fact_954_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ? [M: nat] : Nb = aa(nat,nat,suc,M) ) ).

% not0_implies_Suc
tff(fact_955_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_956_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ~ ! [N: nat] : Xb != aa(nat,nat,suc,N) ) ).

% list_decode.cases
tff(fact_957_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,Uw2: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf((Uu2),(Uv2)),Uw2) ).

% VEBT_internal.membermima.simps(1)
tff(fact_958_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_959_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_960_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_961_not__less0,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% not_less0
tff(fact_962_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),zero_zero(nat)) ).

% less_zeroE
tff(fact_963_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_964_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_965_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P: fun(A,$o),Xb: A] :
      ( ! [X4: A] :
          ( ( aa(A,nat,V2,X4) = zero_zero(nat) )
         => aa(A,$o,P,X4) )
     => ( ! [X4: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V2,X4))
           => ( ~ aa(A,$o,P,X4)
             => ? [Y4: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y4)),aa(A,nat,V2,X4))
                  & ~ aa(A,$o,P,Y4) ) ) )
       => aa(A,$o,P,Xb) ) ) ).

% infinite_descent0_measure
tff(fact_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb) )
    <=> ( ( K2 = zero_zero(nat) )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_975_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_976_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xa: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa),Xa,X) ) ).

% max_def_raw
tff(fact_977_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_978_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_979_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_980_finite__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => aa(set(list(A)),$o,finite_finite(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_af(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) ) ).

% finite_lists_length_eq
tff(fact_981_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_982_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_983_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A),Xb: A,I: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A5)
     => ( member(A,Xb,A5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xb))),A5) ) ) ).

% set_update_subsetI
tff(fact_984_finite__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => aa(set(list(A)),$o,finite_finite(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ag(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) ) ).

% finite_lists_length_le
tff(fact_985_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_986_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_987_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_right_mono
tff(fact_996_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_right_mono_neg
tff(fact_997_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_left_mono
tff(fact_998_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_999_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_1000_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_1001_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_1002_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1003_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_mono
tff(fact_1004_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_1005_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_1006_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_1007_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_1008_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_1009_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ) ) ).

% add_decreasing
tff(fact_1010_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_increasing
tff(fact_1011_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
         => ( aa(A,$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),C3)),B2) ) ) ) ).

% add_decreasing2
tff(fact_1012_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( aa(A,$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),C3)) ) ) ) ).

% add_increasing2
tff(fact_1013_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_1014_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_1015_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_1016_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_1017_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu2: $o,B2: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu2),(B2)),zero_zero(nat)) = $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ).

% vebt_succ.simps(1)
tff(fact_1018_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1019_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1020_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_strict_right_mono
tff(fact_1021_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1022_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1023_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_strict_left_mono
tff(fact_1024_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1025_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1026_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% pos_add_strict
tff(fact_1043_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)
         => ~ ! [C2: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
               => ( C2 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1044_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_1045_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_1046_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_1047_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,B2)),zero_zero(A))
        <=> ( ( aa(A,$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_1048_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% divide_right_mono
tff(fact_1049_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,B2))
        <=> ( ( aa(A,$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_1050_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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1051_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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1052_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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1053_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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1054_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),divide_divide(A,A2,C3)) ) ) ) ).

% divide_right_mono_neg
tff(fact_1055_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_1056_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1057_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% divide_strict_right_mono
tff(fact_1058_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A2,B2))
        <=> ( ( aa(A,$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_1059_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( C3 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_1060_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A2,B2)),zero_zero(A))
        <=> ( ( aa(A,$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_1061_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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_pos_pos
tff(fact_1062_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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_1063_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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_1064_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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_neg_neg
tff(fact_1065_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_1066_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_1067_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_1068_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_1069_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( divide_divide(A,Xb,Y) = 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_1070_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( divide_divide(A,B2,C3) = A2 )
        <=> $ite(C3 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_1071_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = divide_divide(A,B2,C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_1072_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) )
           => ( divide_divide(A,B2,C3) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_1073_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = B2 )
           => ( A2 = divide_divide(A,B2,C3) ) ) ) ) ).

% eq_divide_imp
tff(fact_1074_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( divide_divide(A,B2,C3) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1075_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( A2 = divide_divide(A,B2,C3) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1076_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( divide_divide(A,A2,B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_1077_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_1078_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_1079_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_1080_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = A2 )
        <=> ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1081_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
            & aa(nat,$o,P,aa(nat,nat,suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_1082_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_1083_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I3) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
           => aa(nat,$o,P,aa(nat,nat,suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_1084_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_1085_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_1086_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_1087_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_1088_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
               => ~ aa(nat,$o,P,I4) )
            & aa(nat,$o,P,K) ) ) ) ).

% ex_least_nat_le
tff(fact_1089_option_Osize_I4_J,axiom,
    ! [A: $tType,X2: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_1090_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ? [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1091_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_1092_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_1093_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K2)) ) ) ).

% mult_less_mono1
tff(fact_1094_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),J)) ) ) ).

% mult_less_mono2
tff(fact_1095_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1096_nat__mult__less__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_1097_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_1098_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: 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_1099_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_1100_nat__power__less__imp__less,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Nb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_1101_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_1102_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_1103_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_1104_mod__eq__0D,axiom,
    ! [Mb: nat,D3: nat] :
      ( ( modulo_modulo(nat,Mb,D3) = zero_zero(nat) )
     => ? [Q2: nat] : Mb = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D3),Q2) ) ).

% mod_eq_0D
tff(fact_1105_vebt__insert_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X4: 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),(B4))),X4)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X4: 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),Ts,S)),X4)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,X4: 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)),Ts,S)),X4)
         => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X4: 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,V3)),TreeList,Summary)),X4)
           => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X4: 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)),X4) ) ) ) ) ).

% vebt_insert.cases
tff(fact_1106_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu: $o,Uv: $o,Uw: 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((Uu),(Uv))),Uw)
     => ( ! [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,X4: 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)),X4)
         => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,X4: 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,V3),TreeList,Vc2)),X4)
           => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT,X4: 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,V3),TreeList,Vd2)),X4) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1107_vebt__succ_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu: $o,B4: $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((Uu),(B4))),zero_zero(nat))
     => ( ! [Uv: $o,Uw: $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((Uv),(Uw))),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)
         => ( ! [V3: 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)),V3),zero_zero(nat),Vc2,Vd2)),Ve)
           => ( ! [V3: 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)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Vi)
             => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X4: 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)),X4) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_1108_vebt__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X4: 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),(B4))),X4)
     => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,X4: 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)),Uu,Uv,Uw)),X4)
       => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,X4: 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)),V3),zero_zero(nat),Uy2,Uz2)),X4)
         => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X4: 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)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X4)
           => ~ ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,X4: 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)),X4) ) ) ) ) ).

% vebt_member.cases
tff(fact_1109_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts2,S2),Xb) = vEBT_Node(Info,zero_zero(nat),Ts2,S2) ).

% vebt_insert.simps(2)
tff(fact_1110_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux: nat] : ~ vEBT_V5719532721284313246member(vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1111_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
       => ( ! [Uu: $o] : Xb != vEBT_Leaf((Uu),$true)
         => ( ! [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw,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) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_1112_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))
     => member(A,Xb,aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_1113_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I,Xb) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_1114_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xb: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xb)),J) = $ite(I = J,Xb,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_1115_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv2),(Uw2)),aa(nat,nat,suc,Nb)) = none(nat) ).

% vebt_succ.simps(2)
tff(fact_1116_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
       => ( ! [Uu: $o] : Xb != vEBT_Leaf((Uu),$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_1117_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1118_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1119_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1120_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1121_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1122_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1123_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1124_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1125_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1126_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1127_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1128_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1129_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1130_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1131_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_1132_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_1133_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_1134_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_1135_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_strict_increasing
tff(fact_1136_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_strict_increasing2
tff(fact_1137_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_1138_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)
           => ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1139_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)),divide_divide(A,A2,B2)) ) ) ) ).

% div_positive
tff(fact_1140_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),divide_divide(A,Xb,Z)),divide_divide(A,Y,W)) ) ) ) ) ) ).

% frac_le
tff(fact_1141_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),divide_divide(A,Xb,Z)),divide_divide(A,Y,W)) ) ) ) ) ) ).

% frac_less
tff(fact_1142_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),divide_divide(A,Xb,Z)),divide_divide(A,Y,W)) ) ) ) ) ) ).

% frac_less2
tff(fact_1143_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_1144_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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_1145_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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_1146_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)),divide_divide(A,Xb,Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_1147_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),divide_divide(A,Xb,Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_1148_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),one_one(A))
         => ( aa(A,$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),C3)),A2) ) ) ) ).

% mult_left_le
tff(fact_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A2,B2),C3) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1158_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_1159_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => ( aa(A,$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),divide_divide(A,C3,A2)),divide_divide(A,C3,B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1160_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => ( aa(A,$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),divide_divide(A,C3,A2)),divide_divide(A,C3,B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1161_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),divide_divide(A,Xb,Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1162_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),divide_divide(A,Xb,Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1163_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_1164_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1165_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1166_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_1167_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_1168_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% divide_less_eq
tff(fact_1169_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B2,A2))
        <=> ( ( aa(A,$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_1170_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,A2)),one_one(A))
        <=> ( ( aa(A,$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_1171_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1172_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_1173_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,W: num] :
          ( ( divide_divide(A,B2,C3) = aa(num,A,numeral_numeral(A),W) )
        <=> $ite(C3 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3),aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1174_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C3: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = divide_divide(A,B2,C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3) = B2,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1175_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),divide_divide(A,A2,Z)),B2) = $ite(Z = zero_zero(A),B2,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_1176_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),divide_divide(A,B2,Z)) = $ite(Z = zero_zero(A),A2,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_1177_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),divide_divide(A,Xb,Y)),divide_divide(A,W,Z)) = 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_1178_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),divide_divide(A,Xb,Y)),Z) = 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_1179_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),divide_divide(A,Xb,Y)) = 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_1180_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),divide_divide(A,Y,Z)) = 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_1181_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),divide_divide(A,Xb,Z)),Y) = 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_1182_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_1183_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_1184_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1185_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(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),divide_divide(A,A2,B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1186_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),divide_divide(A,B2,Z)) = $ite(Z = zero_zero(A),A2,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_1187_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),divide_divide(A,Xb,Y)),divide_divide(A,W,Z)) = 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_1188_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),divide_divide(A,Y,Z)) = 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_1189_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),divide_divide(A,Xb,Z)),Y) = 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_1190_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_1191_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_1192_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q3: 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,Q3))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1193_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_1194_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_1195_num_Osize_I5_J,axiom,
    ! [X2: num] : aa(num,nat,size_size(num),aa(num,num,bit0,X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_1196_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),Nb)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),K)
               => ~ aa(nat,$o,P,I4) )
            & aa(nat,$o,P,aa(nat,nat,suc,K)) ) ) ) ).

% ex_least_nat_less
tff(fact_1197_diff__Suc__less,axiom,
    ! [Nb: nat,I: 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,I))),Nb) ) ).

% diff_Suc_less
tff(fact_1198_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_1199_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_1200_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_1201_length__pos__if__in__set,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$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_1202_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_1203_nat__mult__le__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_1204_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,$o,P,zero_zero(nat)) )
          | ? [D5: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
              & ~ aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_1205_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,$o,P,zero_zero(nat)) )
        & ! [D5: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
           => aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split
tff(fact_1206_power__gt__expt,axiom,
    ! [Nb: nat,K2: 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),K2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),K2)) ) ).

% power_gt_expt
tff(fact_1207_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K2: 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),divide_divide(nat,K2,Nb)),divide_divide(nat,K2,Mb)) ) ) ).

% div_le_mono2
tff(fact_1208_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(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_1209_nat__one__le__power,axiom,
    ! [I: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I)
     => 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),I),Nb)) ) ).

% nat_one_le_power
tff(fact_1210_div__less__iff__less__mult,axiom,
    ! [Q3: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,Mb,Q3)),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),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_1211_nat__mult__div__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)) = divide_divide(nat,Mb,Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_1212_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( ( divide_divide(nat,Mb,Nb) = Mb )
      <=> ( Nb = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1213_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),divide_divide(nat,Mb,Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_1214_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_1215_div__less__mono,axiom,
    ! [A5: nat,B5: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A5),B5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( modulo_modulo(nat,A5,Nb) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B5,Nb) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,A5,Nb)),divide_divide(nat,B5,Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_1216_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2),Xb) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) ).

% vebt_insert.simps(3)
tff(fact_1217_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_1218_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_1219_vebt__mint_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A4: $o,B4: $o] : Xb != vEBT_Leaf((A4),(B4))
     => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
       => ~ ! [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) ) ) ).

% vebt_mint.cases
tff(fact_1220_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv: $o] : Xb = vEBT_Leaf($true,(Uv))
           => (Y) )
         => ( ( ? [Uu: $o] : Xb = vEBT_Leaf((Uu),$true)
             => (Y) )
           => ( ( ? [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw,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_1221_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1222_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1223_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1224_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1225_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1226_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1227_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1228_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1229_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),Xb)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1230_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_1231_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1232_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => ( aa(A,$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),divide_divide(A,C3,A2)),divide_divide(A,C3,B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_1233_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_1234_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1235_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1236_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),divide_divide(A,B2,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_1237_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),divide_divide(A,Xb,Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1238_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),divide_divide(A,Xb,Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1239_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => ( aa(A,$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),divide_divide(A,C3,A2)),divide_divide(A,C3,B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1240_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A))
        <=> ( ( aa(A,$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_1241_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2))
        <=> ( ( aa(A,$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_1242_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_1243_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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)),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3)),B2),aa(A,$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_1244_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W)),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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)),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3)),aa(A,$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_1245_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,$o),ord_less_eq(A),divide_divide(A,Xb,Y)),divide_divide(A,W,Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(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_1246_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,$o),ord_less(A),divide_divide(A,Xb,Y)),divide_divide(A,W,Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(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_1247_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_1248_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_1249_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_1250_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),N3)
         => ( aa(A,$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),N3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1251_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N3: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$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),N3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_decreasing
tff(fact_1252_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_1253_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (A4),
                  aa(nat,option(nat),some(nat),zero_zero(nat)),
                  $ite((B4),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( 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_1254_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (B4),
                  aa(nat,option(nat),some(nat),one_one(nat)),
                  $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( 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_1255_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_1256_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_1257_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B4)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts,S) )
             => ( Y != vEBT_Node(Info2,zero_zero(nat),Ts,S) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
               => ( Y != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) ) )
           => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),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,V3)),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,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,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_1258_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_1259_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_1260_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)) = 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_1261_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_1262_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_1263_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)
       => ( divide_divide(nat,Mb,Nb) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ) ) ) ).

% div_geq
tff(fact_1264_div__if,axiom,
    ! [Mb: nat,Nb: 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,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ) ).

% div_if
tff(fact_1265_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_1266_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),divide_divide(nat,Nb,Q3))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q3)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1267_split__div,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,divide_divide(nat,Mb,Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I3)),J3) )
               => aa(nat,$o,P,I3) ) ) ) ) ) ).

% split_div
tff(fact_1268_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),divide_divide(nat,Mb,Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_1269_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),divide_divide(nat,Mb,Nb)))) ) ).

% dividend_less_times_div
tff(fact_1270_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_1271_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,$o,P,Mb) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Nb)
             => ( ( Mb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I3)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_1272_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_1273_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_1274_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_1275_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_1276_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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)),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3)),B2),aa(A,$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_1277_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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)),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C3)),aa(A,$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_1278_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)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ).

% half_gt_zero
tff(fact_1279_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_1280_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R2: A,S2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U)),S2))),V) ) ) ) ) ).

% scaling_mono
tff(fact_1281_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_1282_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_1283_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_1284_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_1285_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = 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,divide_divide(A,A2,B2),C3))),modulo_modulo(A,A2,B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_1286_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_1287_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_1288_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_1289_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( ( A2 != zero_zero(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)),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_1290_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_1291_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_1292_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_1293_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_1294_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_1295_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)
       => ( divide_divide(nat,Mb,Nb) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb)) ) ) ) ).

% le_div_geq
tff(fact_1296_split__div_H,axiom,
    ! [P: fun(nat,$o),Mb: nat,Nb: nat] :
      ( aa(nat,$o,P,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_1297_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_1298_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_1299_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_1300_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_1301_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_1302_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_1303_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_1304_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_1305_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: 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_1306_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_1307_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_1308_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)),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% div_2_gt_zero
tff(fact_1309_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)),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_1310_verit__le__mono__div,axiom,
    ! [A5: nat,B5: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A5),B5)
     => ( 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),divide_divide(nat,A5,Nb)),
                $ite(modulo_modulo(nat,B5,Nb) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            divide_divide(nat,B5,Nb)) ) ) ).

% verit_le_mono_div
tff(fact_1311_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(Uu,zero_zero(nat),Uv,Uw)
           => (Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_1312_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_1313_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb != vEBT_Node(Uu,zero_zero(nat),Uv,Uw)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_1314_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))),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))) = divide_divide(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1315_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_1316_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_1317_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,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,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_1318_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_1319_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_1320_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B4),$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,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,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_1321_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Uu: $o,Uv: $o] : Xb != vEBT_Leaf((Uu),(Uv))
       => ( ! [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,V3: 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,V3),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) )
             => ~ ! [V3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd2)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_1322_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => (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,V3: 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,V3),TreeList,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) )
             => ~ ! [V3: nat,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_1323_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_1324_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_1325_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => (Y) )
         => ( ( ? [V3: 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)),V3),zero_zero(nat),Uy2,Uz2)
             => (Y) )
           => ( ( ? [V3: 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)),V3),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,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,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_1326_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
         => ( ! [V3: 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)),V3),zero_zero(nat),Uy2,Uz2)
           => ( ! [V3: 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)),V3),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,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,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_1327_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),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_1328_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: $o,B4: $o] : A1 = vEBT_Leaf((A4),(B4))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => vEBT_invar_vebt(X,N) )
                 => ( vEBT_invar_vebt(Summary,M)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
                     => ( ( M = N )
                       => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                         => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
                           => ~ ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => vEBT_invar_vebt(X,N) )
                   => ( vEBT_invar_vebt(Summary,M)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
                       => ( ( M = aa(nat,nat,suc,N) )
                         => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                           => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
                             => ~ ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_invar_vebt(X,N) )
                     => ( vEBT_invar_vebt(Summary,M)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
                         => ( ( M = N )
                           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
                             => ( ! [I4: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))
                                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_12)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                                 => ( 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 )
                                         => ! [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))),M))
                                             => ( ( ( vEBT_VEBT_high(Ma,N) = I4 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma,N)) )
                                                & ! [X: nat] :
                                                    ( ( ( vEBT_VEBT_high(X,N) = I4 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X,N)) )
                                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),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 )
                     => ( ! [X: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_invar_vebt(X,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) )
                               => ( ! [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))),M))
                                     => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_12)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                         => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I4: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))
                                               => ( ( ( vEBT_VEBT_high(Ma,N) = I4 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma,N)) )
                                                  & ! [X: nat] :
                                                      ( ( ( vEBT_VEBT_high(X,N) = I4 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X,N)) )
                                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_1329_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A6: $o,B6: $o] : A1 = vEBT_Leaf((A6),(B6))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X3,N4) )
            & vEBT_invar_vebt(Summary2,N4)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X3: vEBT_VEBT] :
                ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X3,N4) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N4))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),aa(nat,nat,suc,N4)) )
            & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_12)
            & ! [X3: vEBT_VEBT] :
                ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
        | ? [TreeList2: list(vEBT_VEBT),N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X3: vEBT_VEBT] :
                ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X3,N4) )
            & vEBT_invar_vebt(Summary2,N4)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),N4) )
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N4))
               => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & 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 )
             => ! [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))),N4))
                 => ( ( ( vEBT_VEBT_high(Ma2,N4) = I3 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma2,N4)) )
                    & ! [X3: nat] :
                        ( ( ( vEBT_VEBT_high(X3,N4) = I3 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X3,N4)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X3)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),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) )
            & ! [X3: vEBT_VEBT] :
                ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X3,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)) )
            & ! [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))),aa(nat,nat,suc,N4)))
               => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I3) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X3: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N4)))
                 => ( ( ( vEBT_VEBT_high(Ma2,N4) = I3 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma2,N4)) )
                    & ! [X3: nat] :
                        ( ( ( vEBT_VEBT_high(X3,N4) = I3 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X3,N4)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X3)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1330_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))),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)) = divide_divide(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1331_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [A5: fun(B,A),B5: 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)),A5),B5),Xb) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A5,Xb)),aa(B,A,B5,Xb)) ) ).

% minus_apply
tff(fact_1332_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_1333_finite__Collect__le__nat,axiom,
    ! [K2: nat] : aa(set(nat),$o,finite_finite(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),K2))) ).

% finite_Collect_le_nat
tff(fact_1334_finite__Collect__less__nat,axiom,
    ! [K2: nat] : aa(set(nat),$o,finite_finite(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),K2))) ).

% finite_Collect_less_nat
tff(fact_1335_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(B),$o,finite_finite(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aj(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),P),Q)))
      <=> ! [Y5: A] :
            ( aa(A,$o,P,Y5)
           => aa(set(B),$o,finite_finite(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_ak(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_1336_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_al(nat,fun(A,$o),Nb))) ) ) ).

% finite_roots_unity
tff(fact_1337_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))
       => ( ! [Uu: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(B4)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = $ite((B4),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((Uu),(B4))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((Uv),(Uw)) )
               => ! [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((Uv),(Uw))),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)) ) )
             => ( ! [V3: 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)),V3),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)),V3),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V3: 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)),V3),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)),V3),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,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,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))),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))),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_1338_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_1339_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_1340_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_1341_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: 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),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1342_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,Nb,K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_1343_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: 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),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_1344_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Nb)
    <=> ( Nb != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_1345_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_1346_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_1347_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% max.bounded_iff
tff(fact_1348_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_1349_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_1350_finite__Diff2,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
      <=> aa(set(A),$o,finite_finite(A),A5) ) ) ).

% finite_Diff2
tff(fact_1351_finite__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) ) ).

% finite_Diff
tff(fact_1352_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_1353_div__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),L)
       => ( divide_divide(int,K2,L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_1354_div__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K2)
       => ( divide_divide(int,K2,L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_1355_mod__neg__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K2)
       => ( modulo_modulo(int,K2,L) = K2 ) ) ) ).

% mod_neg_neg_trivial
tff(fact_1356_mod__pos__pos__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),L)
       => ( modulo_modulo(int,K2,L) = K2 ) ) ) ).

% mod_pos_pos_trivial
tff(fact_1357_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,K2,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),K2),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_1358_half__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),divide_divide(int,K2,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)),K2) ) ).

% half_nonnegative_int_iff
tff(fact_1359_set__bit__greater__eq,axiom,
    ! [K2: int,Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K2)) ).

% set_bit_greater_eq
tff(fact_1360_unset__bit__less__eq,axiom,
    ! [Nb: nat,K2: 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),K2)),K2) ).

% unset_bit_less_eq
tff(fact_1361_int__div__less__self,axiom,
    ! [Xb: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),K2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,Xb,K2)),Xb) ) ) ).

% int_div_less_self
tff(fact_1362_neg__mod__bound,axiom,
    ! [L: int,K2: 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,K2,L)) ) ).

% neg_mod_bound
tff(fact_1363_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K2: 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,K2,L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_1364_neg__mod__sign,axiom,
    ! [L: int,K2: 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,K2,L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_1365_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K2: 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,K2,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_1366_mod__pos__geq,axiom,
    ! [L: int,K2: 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),K2)
       => ( modulo_modulo(int,K2,L) = modulo_modulo(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1367_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_1368_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Nb),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_1369_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_1370_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Nb),zero_zero(extended_enat))
    <=> ( Nb = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_1371_i0__lb,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),zero_zero(extended_enat)),Nb) ).

% i0_lb
tff(fact_1372_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) )
    <=> ! [X3: nat] :
          ( member(nat,X3,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X3) ) ) ).

% all_nat_less
tff(fact_1373_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) )
    <=> ? [X3: nat] :
          ( member(nat,X3,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X3) ) ) ).

% ex_nat_less
tff(fact_1374_finite__maxlen,axiom,
    ! [A: $tType,M6: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite(list(A)),M6)
     => ? [N: nat] :
        ! [X: list(A)] :
          ( member(list(A),X,M6)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),N) ) ) ).

% finite_maxlen
tff(fact_1375_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R3: int,Q3: int,R2: 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)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q3),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1376_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R3: int,Q3: int,R2: 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)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1377_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q3: int,R2: int,B3: int,Q5: int,R3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R3) )
     => ( 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),B3),Q5)),R3)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1378_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q3: int,R2: int,B3: int,Q5: int,R3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R3) )
     => ( 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),B3),Q5)),R3))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B3)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q3),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1379_q__pos__lemma,axiom,
    ! [B3: int,Q5: int,R3: 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),B3),Q5)),R3))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R3),B3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_1380_zdiv__zmult2__eq,axiom,
    ! [C3: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C3)
     => ( divide_divide(int,A2,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) = divide_divide(int,divide_divide(int,A2,B2),C3) ) ) ).

% zdiv_zmult2_eq
tff(fact_1381_div__pos__geq,axiom,
    ! [L: int,K2: 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),K2)
       => ( divide_divide(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),L),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1382_zmod__eq__0__iff,axiom,
    ! [Mb: int,D3: int] :
      ( ( modulo_modulo(int,Mb,D3) = zero_zero(int) )
    <=> ? [Q4: int] : Mb = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q4) ) ).

% zmod_eq_0_iff
tff(fact_1383_zmod__eq__0D,axiom,
    ! [Mb: int,D3: int] :
      ( ( modulo_modulo(int,Mb,D3) = zero_zero(int) )
     => ? [Q2: int] : Mb = aa(int,int,aa(int,fun(int,int),times_times(int),D3),Q2) ) ).

% zmod_eq_0D
tff(fact_1384_mod__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)),zero_zero(int))
       => ( modulo_modulo(int,K2,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_1385_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_1386_realpow__pos__nth2,axiom,
    ! [A2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ? [R4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R4),aa(nat,nat,suc,Nb)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1387_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_1388_split__zdiv,axiom,
    ! [P: fun(int,$o),Nb: int,K2: int] :
      ( aa(int,$o,P,divide_divide(int,Nb,K2))
    <=> ( ( ( K2 = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K2)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I3)),J3) ) )
             => aa(int,$o,P,I3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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),K2),I3)),J3) ) )
             => aa(int,$o,P,I3) ) ) ) ) ).

% split_zdiv
tff(fact_1389_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( divide_divide(int,A2,B2) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1390_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( divide_divide(int,A2,B2) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1391_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_1392_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( modulo_modulo(int,A2,B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_1393_split__zmod,axiom,
    ! [P: fun(int,$o),Nb: int,K2: int] :
      ( aa(int,$o,P,modulo_modulo(int,Nb,K2))
    <=> ( ( ( K2 = zero_zero(int) )
         => aa(int,$o,P,Nb) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K2)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I3)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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),K2),I3)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_1394_verit__le__mono__div__int,axiom,
    ! [A5: int,B5: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A5),B5)
     => ( 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),divide_divide(int,A5,Nb)),
                $ite(modulo_modulo(int,B5,Nb) = zero_zero(int),one_one(int),zero_zero(int)))),
            divide_divide(int,B5,Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_1395_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)
       => ? [R4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R4),Nb) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1396_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)
       => ? [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X4)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X4),Nb) = A2 )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y4)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),Nb) = A2 ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1397_split__neg__lemma,axiom,
    ! [K2: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,Nb,K2)),modulo_modulo(int,Nb,K2))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1398_split__pos__lemma,axiom,
    ! [K2: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,Nb,K2)),modulo_modulo(int,Nb,K2))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K2)
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1399_zmod__zmult2__eq,axiom,
    ! [C3: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C3)
     => ( modulo_modulo(int,A2,aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) = 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,divide_divide(int,A2,B2),C3))),modulo_modulo(int,A2,B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1400_int__power__div__base,axiom,
    ! [Mb: nat,K2: 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)),K2)
       => ( divide_divide(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),K2),Mb),K2) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K2),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_1401_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,A2,A5)
           => ? [X4: A] :
                ( member(A,X4,A5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A2)
                & ! [Xa: A] :
                    ( member(A,Xa,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X4)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_1402_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,A2,A5)
           => ? [X4: A] :
                ( member(A,X4,A5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4)
                & ! [Xa: A] :
                    ( member(A,Xa,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_1403_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))
     => ( 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)) = 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_1404_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)
     => ( 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)) = divide_divide(int,B2,A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_1405_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_1406_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_1407_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A2: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),D3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ) ).

% max.mono
tff(fact_1408_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_1409_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_1410_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% max.boundedE
tff(fact_1411_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2) ) ) ) ).

% max.boundedI
tff(fact_1412_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_1413_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_1414_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_1415_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_1416_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_1417_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_1418_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI1
tff(fact_1419_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI2
tff(fact_1420_Diff__infinite__finite,axiom,
    ! [A: $tType,T2: set(A),S3: set(A)] :
      ( aa(set(A),$o,finite_finite(A),T2)
     => ( ~ aa(set(A),$o,finite_finite(A),S3)
       => ~ aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T2)) ) ) ).

% Diff_infinite_finite
tff(fact_1421_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_1422_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% max.strict_boundedE
tff(fact_1423_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_1424_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_1425_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_1426_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),P))
     => aa(set(B),$o,finite_finite(B),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,B),fun(B,$o)),P),F3))) ) ).

% finite_image_set
tff(fact_1427_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,$o),F3: fun(A,fun(B,C))] :
      ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(B),$o,finite_finite(B),aa(fun(B,$o),set(B),collect(B),Q))
       => aa(set(C),$o,finite_finite(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_an(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),P),Q),F3))) ) ) ).

% finite_image_set2
tff(fact_1428_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A5: fun(A,B),B5: fun(A,B),X: 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)),A5),B5),X) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A5,X)),aa(A,B,B5,X)) ) ).

% fun_diff_def
tff(fact_1429_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,A5)
                & ! [Xa: A] :
                    ( member(A,Xa,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X4)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_1430_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,A5)
                & ! [Xa: A] :
                    ( member(A,Xa,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa)
                     => ( X4 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_1431_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,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B4)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) )
               => ~ 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),(B4))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts,S) )
               => ( ( Y = vEBT_Node(Info2,zero_zero(nat),Ts,S) )
                 => ~ 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),Ts,S)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
                 => ( ( Y = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) )
                   => ~ 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)),Ts,S)),Xaa)) ) )
             => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),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,V3)),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,V3)),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,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,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_1432_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,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( 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),(B4))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ~ 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)),Uu,Uv,Uw)),Xaa)) )
           => ( ! [V3: 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)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xaa)) )
             => ( ! [V3: 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)),V3),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)),V3),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,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,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_1433_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,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$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),(B4))),Xaa)) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ~ (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)),Uu,Uv,Uw)),Xaa)) ) )
           => ( ! [V3: 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)),V3),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)),V3),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: 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)),V3),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)),V3),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,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,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_1434_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,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( 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),(B4))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu,zero_zero(nat),Uv,Uw) )
               => ~ 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(Uu,zero_zero(nat),Uv,Uw)),Xaa)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_1435_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,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( 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),(B4))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_1436_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,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$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),(B4))),Xaa)) ) )
         => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu,zero_zero(nat),Uv,Uw) )
               => ( ~ (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(Uu,zero_zero(nat),Uv,Uw)),Xaa)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,V3),TreeList,S)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1437_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,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( 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),(B4))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$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,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,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_1438_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))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ~ 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((Uu),(Uv))),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,V3: 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,V3),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,V3),TreeList,Vc2)),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) )
               => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),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,V3),TreeList,Vd2)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_1439_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Q3),zero_zero(extended_enat)) = Q3 ).

% max_enat_simps(2)
tff(fact_1440_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),zero_zero(extended_enat)),Q3) = Q3 ).

% max_enat_simps(3)
tff(fact_1441_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: 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),K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_1442_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: 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),K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_1443_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Nb,K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_1444_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_1445_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_1446_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,V3: 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,V3),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,V3),TreeList,Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) )
           => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),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,V3),TreeList,Vd2)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_1447_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))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( ~ (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((Uu),(Uv))),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,V3: 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,V3),TreeList,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,V3),TreeList,Vc2)),Xaa)) ) )
               => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),$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,V3),TreeList,Vd2)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_1448_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_1449_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_1450_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite(A),set_or1337092689740270186AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_1451_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_1452_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1453_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_1454_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_1455_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1456_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_1457_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_1458_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_1459_bounded__Max__nat,axiom,
    ! [P: fun(nat,$o),Xb: nat,M6: nat] :
      ( aa(nat,$o,P,Xb)
     => ( ! [X4: nat] :
            ( aa(nat,$o,P,X4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),M6) )
       => ~ ! [M: nat] :
              ( aa(nat,$o,P,M)
             => ~ ! [X: nat] :
                    ( aa(nat,$o,P,X)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),M) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1460_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F2: fun(nat,fun(A,A)),A4: nat,B4: 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))),F2),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),B4),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1461_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_1462_times__int__code_I1_J,axiom,
    ! [K2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K2),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_1463_minus__int__code_I1_J,axiom,
    ! [K2: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),zero_zero(int)) = K2 ).

% minus_int_code(1)
tff(fact_1464_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_1465_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_1466_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_1467_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_1468_finite__nat__set__iff__bounded,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),N3)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( member(nat,X3,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_1469_bounded__nat__set__is__finite,axiom,
    ! [N3: set(nat),Nb: nat] :
      ( ! [X4: nat] :
          ( member(nat,X4,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),Nb) )
     => aa(set(nat),$o,finite_finite(nat),N3) ) ).

% bounded_nat_set_is_finite
tff(fact_1470_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),N3)
    <=> ? [M5: nat] :
        ! [X3: nat] :
          ( member(nat,X3,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_1471_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ao(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_1472_finite__less__ub,axiom,
    ! [F3: fun(nat,nat),U: nat] :
      ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,F3,N))
     => aa(set(nat),$o,finite_finite(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ap(fun(nat,nat),fun(nat,fun(nat,$o)),F3),U))) ) ).

% finite_less_ub
tff(fact_1473_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite(A),set_or1337092689740270186AtMost(A,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_1474_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K2),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1475_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_1476_int__ge__induct,axiom,
    ! [K2: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),I)
     => ( aa(int,$o,P,K2)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_1477_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_1478_int__gr__induct,axiom,
    ! [K2: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),I)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_1479_int__le__induct,axiom,
    ! [I: int,K2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K2)
     => ( aa(int,$o,P,K2)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_1480_int__less__induct,axiom,
    ! [I: int,K2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K2)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_1481_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_1482_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_1483_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_1484_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_1485_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_1486_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_1487_int__induct,axiom,
    ! [P: fun(int,$o),K2: int,I: int] :
      ( aa(int,$o,P,K2)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),I2)
           => ( aa(int,$o,P,I2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_1488_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_1489_cppi,axiom,
    ! [D4: int,P: fun(int,$o),P3: fun(int,$o),A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X4)
           => ( aa(int,$o,P,X4)
            <=> aa(int,$o,P3,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,A5)
                     => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X4)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)) ) )
         => ( ! [X4: int,K: int] :
                ( aa(int,$o,P3,X4)
              <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D4))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X3: int] :
                    ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & aa(int,$o,P3,X3) )
                | ? [X3: int] :
                    ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & ? [Xa2: int] :
                        ( member(int,Xa2,A5)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa2),X3)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_1490_cpmi,axiom,
    ! [D4: int,P: fun(int,$o),P3: fun(int,$o),B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Z4)
           => ( aa(int,$o,P,X4)
            <=> aa(int,$o,P3,X4) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,B5)
                     => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X4)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4)) ) )
         => ( ! [X4: int,K: int] :
                ( aa(int,$o,P3,X4)
              <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D4))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X3: int] :
                    ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & aa(int,$o,P3,X3) )
                | ? [X3: int] :
                    ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D4))
                    & ? [Xa2: int] :
                        ( member(int,Xa2,B5)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X3)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_1491_aset_I8_J,axiom,
    ! [D4: int,A5: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A5)
                 => ( X != 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),X)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) ) ) ).

% aset(8)
tff(fact_1492_aset_I6_J,axiom,
    ! [D4: int,Ta: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != 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),X),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_1493_bset_I8_J,axiom,
    ! [D4: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int)),B5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != 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),X)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)) ) ) ) ) ).

% bset(8)
tff(fact_1494_bset_I6_J,axiom,
    ! [D4: int,B5: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X != 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),X),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)),Ta) ) ) ) ).

% bset(6)
tff(fact_1495_finite__nth__roots,axiom,
    ! [Nb: nat,C3: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => aa(set(complex),$o,finite_finite(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_aq(nat,fun(complex,fun(complex,$o)),Nb),C3))) ) ).

% finite_nth_roots
tff(fact_1496_aset_I7_J,axiom,
    ! [D4: int,A5: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A5)
                 => ( X != 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),X)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) ) ) ).

% aset(7)
tff(fact_1497_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_1498_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X) ) ) ).

% minf(7)
tff(fact_1499_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Ta) ) ) ).

% minf(5)
tff(fact_1500_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
         => ( X != Ta ) ) ) ).

% minf(4)
tff(fact_1501_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
         => ( X != Ta ) ) ) ).

% minf(3)
tff(fact_1502_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P3,X4) ) )
         => ( ? [Z4: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q6,X4) ) )
           => ? [Z3: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
               => ( ( aa(A,$o,P,X)
                    | aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P3,X)
                    | aa(A,$o,Q6,X) ) ) ) ) ) ) ).

% minf(2)
tff(fact_1503_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P3,X4) ) )
         => ( ? [Z4: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z4)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q6,X4) ) )
           => ? [Z3: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
               => ( ( aa(A,$o,P,X)
                    & aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P3,X)
                    & aa(A,$o,Q6,X) ) ) ) ) ) ) ).

% minf(1)
tff(fact_1504_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_1505_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X) ) ) ).

% pinf(7)
tff(fact_1506_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Ta) ) ) ).

% pinf(5)
tff(fact_1507_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
         => ( X != Ta ) ) ) ).

% pinf(4)
tff(fact_1508_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
         => ( X != Ta ) ) ) ).

% pinf(3)
tff(fact_1509_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P3,X4) ) )
         => ( ? [Z4: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q6,X4) ) )
           => ? [Z3: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
               => ( ( aa(A,$o,P,X)
                    | aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P3,X)
                    | aa(A,$o,Q6,X) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_1510_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
             => ( aa(A,$o,P,X4)
              <=> aa(A,$o,P3,X4) ) )
         => ( ? [Z4: A] :
              ! [X4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X4)
               => ( aa(A,$o,Q,X4)
                <=> aa(A,$o,Q6,X4) ) )
           => ? [Z3: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
               => ( ( aa(A,$o,P,X)
                    & aa(A,$o,Q,X) )
                <=> ( aa(A,$o,P3,X)
                    & aa(A,$o,Q6,X) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_1511_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X) ) ) ).

% minf(8)
tff(fact_1512_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Ta) ) ) ).

% minf(6)
tff(fact_1513_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X) ) ) ).

% pinf(8)
tff(fact_1514_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Ta) ) ) ).

% pinf(6)
tff(fact_1515_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
          ( ! [X4: A,K: A] :
              ( aa(A,$o,P,X4)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K),D4))) )
         => ( ! [X4: A,K: A] :
                ( aa(A,$o,Q,X4)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K),D4))) )
           => ! [X: A,K4: A] :
                ( ( aa(A,$o,P,X)
                  | aa(A,$o,Q,X) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4)))
                  | aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1516_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D4: A,Q: fun(A,$o)] :
          ( ! [X4: A,K: A] :
              ( aa(A,$o,P,X4)
            <=> aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K),D4))) )
         => ( ! [X4: A,K: A] :
                ( aa(A,$o,Q,X4)
              <=> aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),aa(A,A,aa(A,fun(A,A),times_times(A),K),D4))) )
           => ! [X: A,K4: A] :
                ( ( aa(A,$o,P,X)
                  & aa(A,$o,Q,X) )
              <=> ( aa(A,$o,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4)))
                  & aa(A,$o,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1517_bset_I1_J,axiom,
    ! [D4: int,B5: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B5)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B5)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4)) ) )
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X)
                & aa(int,$o,Q,X) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)) ) ) ) ) ) ).

% bset(1)
tff(fact_1518_bset_I2_J,axiom,
    ! [D4: int,B5: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B5)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B5)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D4)) ) )
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X)
                | aa(int,$o,Q,X) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)) ) ) ) ) ) ).

% bset(2)
tff(fact_1519_aset_I1_J,axiom,
    ! [D4: int,A5: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A5)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A5)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)) ) )
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X)
                & aa(int,$o,Q,X) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) ) ) ) ) ).

% aset(1)
tff(fact_1520_aset_I2_J,axiom,
    ! [D4: int,A5: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X4: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A5)
                 => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)) ) )
     => ( ! [X4: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A5)
                   => ( X4 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X4)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D4)) ) )
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( aa(int,$o,P,X)
                | aa(int,$o,Q,X) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)) ) ) ) ) ) ).

% aset(2)
tff(fact_1521_minusinfinity,axiom,
    ! [D3: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int,K: int] :
            ( aa(int,$o,P1,X4)
          <=> aa(int,$o,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X4),Z4)
             => ( aa(int,$o,P,X4)
              <=> aa(int,$o,P1,X4) ) )
         => ( ? [X_1: int] : aa(int,$o,P1,X_1)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_1522_plusinfinity,axiom,
    ! [D3: int,P3: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int,K: int] :
            ( aa(int,$o,P3,X4)
          <=> aa(int,$o,P3,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X4)
             => ( aa(int,$o,P,X4)
              <=> aa(int,$o,P3,X4) ) )
         => ( ? [X_1: int] : aa(int,$o,P3,X_1)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_1523_incr__mult__lemma,axiom,
    ! [D3: int,P: fun(int,$o),K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int] :
            ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
         => ! [X: int] :
              ( aa(int,$o,P,X)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1524_decr__mult__lemma,axiom,
    ! [D3: int,P: fun(int,$o),K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int] :
            ( aa(int,$o,P,X4)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),D3)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
         => ! [X: int] :
              ( aa(int,$o,P,X)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1525_periodic__finite__ex,axiom,
    ! [D3: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ! [X4: int,K: int] :
            ( aa(int,$o,P,X4)
          <=> aa(int,$o,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X4),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3))) )
       => ( ? [X_12: int] : aa(int,$o,P,X_12)
        <=> ? [X3: int] :
              ( member(int,X3,set_or1337092689740270186AtMost(int,one_one(int),D3))
              & aa(int,$o,P,X3) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1526_bset_I3_J,axiom,
    ! [D4: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Ta),one_one(int)),B5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X = Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_1527_bset_I4_J,axiom,
    ! [D4: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,Ta,B5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
           => ( ( X != Ta )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_1528_bset_I5_J,axiom,
    ! [D4: int,B5: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)),Ta) ) ) ) ).

% bset(5)
tff(fact_1529_bset_I7_J,axiom,
    ! [D4: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,Ta,B5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X != 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),X)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)) ) ) ) ) ).

% bset(7)
tff(fact_1530_aset_I3_J,axiom,
    ! [D4: int,Ta: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_1531_aset_I4_J,axiom,
    ! [D4: int,Ta: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,Ta,A5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( ( X != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_1532_aset_I5_J,axiom,
    ! [D4: int,Ta: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D4)
     => ( member(int,Ta,A5)
       => ! [X: int] :
            ( ! [Xa3: int] :
                ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_1533_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [A4: real,B4: real,C2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A4),B4)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B4),C2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),B4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B4),C2)
                 => aa(real,$o,aa(real,fun(real,$o),P,A4),C2) ) ) ) )
       => ( ! [X4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2)
               => ? [D6: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                    & ! [A4: real,B4: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),X4)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B4)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B4),A4)),D6) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A4),B4) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_1534_Suc__if__eq,axiom,
    ! [A: $tType,F3: fun(nat,A),H: fun(nat,A),G3: A,Nb: nat] :
      ( ! [N: nat] : aa(nat,A,F3,aa(nat,nat,suc,N)) = aa(nat,A,H,N)
     => ( ( aa(nat,A,F3,zero_zero(nat)) = G3 )
       => ( aa(nat,A,F3,Nb) = $ite(Nb = zero_zero(nat),G3,aa(nat,A,H,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_1535_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_1536_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_1537_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_1538_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R2: 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),Q3),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),Q3),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))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_1539_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ar(set(A),fun(fun(A,B),fun(A,$o)),I6),Xb)))
         => ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ar(set(A),fun(fun(A,B),fun(A,$o)),I6),Y)))
           => aa(set(A),$o,finite_finite(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,$o)),aTP_Lamp_as(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I6),Xb),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_1540_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q3: int] :
      ( ( L != zero_zero(int) )
     => ( ( K2 = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L) )
       => eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_1541_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
    <=> ( ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R2) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
              Q3 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1542_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_1543_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R2: 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),Q3),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),Q3),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))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_1544_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_at(set(A),fun(fun(A,B),fun(A,$o)),I6),Xb)))
         => ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_at(set(A),fun(fun(A,B),fun(A,$o)),I6),Y)))
           => aa(set(A),$o,finite_finite(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,$o)),aTP_Lamp_au(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I6),Xb),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_1545_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_1546_concat__bit__Suc,axiom,
    ! [Nb: nat,K2: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,Nb),K2),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K2,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,divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),L))) ).

% concat_bit_Suc
tff(fact_1547_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_1548_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K2: A,F3: fun(A,nat),Nb: nat] :
      ( aa(A,$o,P,K2)
     => ( ! [X4: A] :
            ( aa(A,$o,P,X4)
           => ? [Y4: A] :
                ( aa(A,$o,P,Y4)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,Y4)),aa(A,nat,F3,X4)) ) )
       => ? [Y3: A] :
            ( aa(A,$o,P,Y3)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,K2)),Nb)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1549_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ( aa(nat,$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_1550_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X2: A] : aa(option(A),nat,size_option(A,Xb),aa(A,option(A),some(A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_1551_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( dvd_dvd(nat,Mb,one_one(nat))
    <=> ( Mb = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_1552_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( dvd_dvd(A,zero_zero(A),A2)
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_1553_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : dvd_dvd(A,A2,zero_zero(A)) ) ).

% dvd_0_right
tff(fact_1554_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1555_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1556_dvd__1__left,axiom,
    ! [K2: nat] : dvd_dvd(nat,aa(nat,nat,suc,zero_zero(nat)),K2) ).

% dvd_1_left
tff(fact_1557_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( 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_1558_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,A2,C3)
           => ( dvd_dvd(A,divide_divide(A,B2,A2),divide_divide(A,C3,A2))
            <=> dvd_dvd(A,B2,C3) ) ) ) ) ).

% div_dvd_div
tff(fact_1559_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
    <=> ( ( K2 = zero_zero(nat) )
        | dvd_dvd(nat,Mb,Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1560_concat__bit__0,axiom,
    ! [K2: int,L: int] : aa(int,int,bit_concat_bit(zero_zero(nat),K2),L) = L ).

% concat_bit_0
tff(fact_1561_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_1562_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: A,A2: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( C3 = zero_zero(A) )
            | dvd_dvd(A,A2,B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_1563_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C3: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( C3 = zero_zero(A) )
            | dvd_dvd(A,A2,B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_1564_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 != zero_zero(A) )
         => ( 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),C3))
          <=> dvd_dvd(A,B2,C3) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1565_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 != zero_zero(A) )
         => ( 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),C3),A2))
          <=> dvd_dvd(A,B2,C3) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1566_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( dvd_dvd(A,B2,one_one(A))
           => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_1567_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( 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),C3),A2)),B2))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1568_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C3: A] :
          ( 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),C3),A2)))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1569_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_1570_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_1571_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( dvd_dvd(A,B2,one_one(A))
           => dvd_dvd(A,divide_divide(A,A2,B2),one_one(A)) ) ) ) ).

% unit_div
tff(fact_1572_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => dvd_dvd(A,divide_divide(A,one_one(A),A2),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_1573_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( divide_divide(A,one_one(A),divide_divide(A,one_one(A),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_1574_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A2: A,B2: A] :
          ( dvd_dvd(A,C3,A2)
         => ( dvd_dvd(A,C3,B2)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ) ).

% div_add
tff(fact_1575_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C3: A,A2: A,B2: A] :
          ( dvd_dvd(A,C3,A2)
         => ( dvd_dvd(A,C3,B2)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ) ).

% div_diff
tff(fact_1576_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1577_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K2: 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,K2),L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_1578_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_concat_bit(Nb,K2),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_1579_dbl__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K2)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,K2)) ) ).

% dbl_simps(5)
tff(fact_1580_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),divide_divide(A,one_one(A),A2)) = divide_divide(A,B2,A2) ) ) ) ).

% unit_mult_div_div
tff(fact_1581_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_1582_even__Suc,axiom,
    ! [Nb: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,Nb))
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb) ) ).

% even_Suc
tff(fact_1583_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,aa(nat,nat,suc,Nb)))
    <=> dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_1584_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)
         => ( 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))
          <=> dvd_dvd(A,A2,B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_1585_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( 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))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_mult_iff
tff(fact_1586_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ 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))
        <=> ~ ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
            <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% odd_add
tff(fact_1587_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( 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))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
          <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_add
tff(fact_1588_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( 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))))
        <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ).

% even_mod_2_iff
tff(fact_1589_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
     => ( divide_divide(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_1590_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
     => ( divide_divide(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ).

% even_Suc_div_two
tff(fact_1591_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1592_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),zero_zero(A))
        <=> ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_1593_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1594_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( 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)))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ).

% even_plus_one_iff
tff(fact_1595_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A,B2: A] :
          ( 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))
        <=> 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_1596_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ 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_1597_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( 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)
        | 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_1598_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
              & ( A2 != zero_zero(A) ) )
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1599_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ( 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))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1600_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ( 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),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1601_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ( 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))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1602_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% even_power
tff(fact_1603_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ 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))),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_1604_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ 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))),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_1605_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,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W))
            & ( ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W))
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( 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_1606_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( 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_1607_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => ( 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)) = 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_1608_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,B2,C3)
           => dvd_dvd(A,A2,C3) ) ) ) ).

% dvd_trans
tff(fact_1609_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : dvd_dvd(A,A2,A2) ) ).

% dvd_refl
tff(fact_1610_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P2: A,A2: A,B2: A] :
          ( dvd_dvd(A,P2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ~ ! [X4: A,Y3: A] :
                ( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X4),Y3) )
               => ( dvd_dvd(A,X4,A2)
                 => ~ dvd_dvd(A,Y3,B2) ) ) ) ) ).

% dvd_productE
tff(fact_1611_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
         => ? [B7: A,C6: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B7),C6) )
              & dvd_dvd(A,B7,B2)
              & dvd_dvd(A,C6,C3) ) ) ) ).

% division_decomp
tff(fact_1612_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( dvd_dvd(A,zero_zero(A),A2)
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1613_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
         => ~ ! [K: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) ) ) ).

% dvdE
tff(fact_1614_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A2: A,B2: A,K2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) )
         => dvd_dvd(A,B2,A2) ) ) ).

% dvdI
tff(fact_1615_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( 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_1616_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C3: A,B2: A] :
          ( dvd_dvd(A,A2,C3)
         => dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ).

% dvd_mult
tff(fact_1617_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,B2)
         => dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ).

% dvd_mult2
tff(fact_1618_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3)
         => dvd_dvd(A,A2,C3) ) ) ).

% dvd_mult_left
tff(fact_1619_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% dvd_triv_left
tff(fact_1620_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,C3,D3)
           => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ).

% mult_dvd_mono
tff(fact_1621_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3)
         => dvd_dvd(A,B2,C3) ) ) ).

% dvd_mult_right
tff(fact_1622_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) ) ).

% dvd_triv_right
tff(fact_1623_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : dvd_dvd(A,one_one(A),A2) ) ).

% one_dvd
tff(fact_1624_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => dvd_dvd(A,B2,A2) ) ) ).

% unit_imp_dvd
tff(fact_1625_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,B2,one_one(A))
           => dvd_dvd(A,A2,one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1626_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,A2,C3)
           => dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ) ) ).

% dvd_add
tff(fact_1627_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( dvd_dvd(A,A2,C3)
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
          <=> dvd_dvd(A,A2,B2) ) ) ) ).

% dvd_add_left_iff
tff(fact_1628_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,B2)
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
          <=> dvd_dvd(A,A2,C3) ) ) ) ).

% dvd_add_right_iff
tff(fact_1629_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( dvd_dvd(A,Xb,Y)
         => ( dvd_dvd(A,Xb,Z)
           => dvd_dvd(A,Xb,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ) ) ).

% dvd_diff
tff(fact_1630_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A2: A,C3: A,B2: A] :
          ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))
        <=> dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) ) ) ).

% dvd_diff_commute
tff(fact_1631_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C3: A,A2: A,B2: A] :
          ( dvd_dvd(A,C3,A2)
         => ( dvd_dvd(A,C3,B2)
           => ( ( divide_divide(A,A2,C3) = divide_divide(A,B2,C3) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1632_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( divide_divide(A,A2,C3) = divide_divide(A,B2,C3) )
         => ( dvd_dvd(A,C3,A2)
           => ( dvd_dvd(A,C3,B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1633_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D3: A,B2: A,A2: A] :
          ( dvd_dvd(A,D3,B2)
         => ( dvd_dvd(A,B2,A2)
           => ( divide_divide(A,divide_divide(A,A2,D3),divide_divide(A,B2,D3)) = divide_divide(A,A2,B2) ) ) ) ) ).

% div_div_div_same
tff(fact_1634_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A,Nb: nat] :
          ( dvd_dvd(A,Xb,Y)
         => 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_1635_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,B2: A,A2: A] :
          ( dvd_dvd(A,C3,B2)
         => ( modulo_modulo(A,modulo_modulo(A,A2,B2),C3) = modulo_modulo(A,A2,C3) ) ) ) ).

% mod_mod_cancel
tff(fact_1636_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K2: A,Mb: A,Nb: A] :
          ( dvd_dvd(A,K2,Mb)
         => ( dvd_dvd(A,K2,Nb)
           => dvd_dvd(A,K2,modulo_modulo(A,Mb,Nb)) ) ) ) ).

% dvd_mod
tff(fact_1637_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C3: A,A2: A,B2: A] :
          ( dvd_dvd(A,C3,modulo_modulo(A,A2,B2))
         => ( dvd_dvd(A,C3,B2)
           => dvd_dvd(A,C3,A2) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1638_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C3: A,B2: A,A2: A] :
          ( dvd_dvd(A,C3,B2)
         => ( dvd_dvd(A,C3,modulo_modulo(A,A2,B2))
          <=> dvd_dvd(A,C3,A2) ) ) ) ).

% dvd_mod_iff
tff(fact_1639_dvd__diff__nat,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,K2,Mb)
     => ( dvd_dvd(nat,K2,Nb)
       => dvd_dvd(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_1640_zdvd__zdiffD,axiom,
    ! [K2: int,Mb: int,Nb: int] :
      ( dvd_dvd(int,K2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Mb),Nb))
     => ( dvd_dvd(int,K2,Nb)
       => dvd_dvd(int,K2,Mb) ) ) ).

% zdvd_zdiffD
tff(fact_1641_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( 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_1642_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D2: nat,X4: nat,Y3: nat] :
      ( dvd_dvd(nat,D2,A2)
      & dvd_dvd(nat,D2,B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D2) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_1643_bezout__lemma__nat,axiom,
    ! [D3: nat,A2: nat,B2: nat,Xb: nat,Y: nat] :
      ( dvd_dvd(nat,D3,A2)
     => ( dvd_dvd(nat,D3,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)),D3) )
            | ( 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)),D3) ) )
         => ? [X4: nat,Y3: nat] :
              ( dvd_dvd(nat,D3,A2)
              & dvd_dvd(nat,D3,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),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Y3)),D3) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),D3) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1644_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D2: nat,X4: nat,Y3: nat] :
      ( dvd_dvd(nat,D2,A2)
      & dvd_dvd(nat,D2,B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = D2 )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_1645_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_av(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_av(A,fun(A,$o),B2)))
        <=> dvd_dvd(A,A2,B2) ) ) ).

% subset_divisors_dvd
tff(fact_1646_concat__bit__assoc,axiom,
    ! [Nb: nat,K2: int,Mb: nat,L: int,R2: int] : aa(int,int,bit_concat_bit(Nb,K2),aa(int,int,bit_concat_bit(Mb,L),R2)) = 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,K2),L)),R2) ).

% concat_bit_assoc
tff(fact_1647_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_av(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_av(A,fun(A,$o),B2)))
        <=> ( dvd_dvd(A,A2,B2)
            & ~ dvd_dvd(A,B2,A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_1648_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ dvd_dvd(A,zero_zero(A),one_one(A)) ) ).

% not_is_unit_0
tff(fact_1649_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S2: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
         => ( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S2))
          <=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S2)) ) ) ) ).

% pinf(9)
tff(fact_1650_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S2: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X)
         => ( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S2))
          <=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S2)) ) ) ) ).

% pinf(10)
tff(fact_1651_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S2: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
         => ( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S2))
          <=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S2)) ) ) ) ).

% minf(9)
tff(fact_1652_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D3: A,S2: A] :
        ? [Z3: A] :
        ! [X: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z3)
         => ( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S2))
          <=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),S2)) ) ) ) ).

% minf(10)
tff(fact_1653_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1654_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),one_one(A))
        <=> ( dvd_dvd(A,A2,one_one(A))
            & dvd_dvd(A,B2,one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_1655_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> dvd_dvd(A,A2,C3) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_1656_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3)
          <=> dvd_dvd(A,A2,C3) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_1657_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
          <=> dvd_dvd(A,A2,C3) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_1658_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3)
          <=> dvd_dvd(A,B2,C3) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_1659_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( 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),C3) )
          <=> ( B2 = C3 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_1660_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( 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),C3),A2) )
          <=> ( B2 = C3 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_1661_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( dvd_dvd(A,C3,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C3)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C3) ) ) ) ).

% dvd_div_mult
tff(fact_1662_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( dvd_dvd(A,C3,B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ) ) ).

% div_mult_swap
tff(fact_1663_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( dvd_dvd(A,C3,B2)
         => ( dvd_dvd(A,B2,A2)
           => ( divide_divide(A,A2,divide_divide(A,B2,C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),C3) ) ) ) ) ).

% div_div_eq_right
tff(fact_1664_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C3: A,A2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3),A2)
         => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A2,B2),C3) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_1665_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A,B2: A] :
          ( dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),B2)
         => dvd_dvd(A,A2,divide_divide(A,B2,C3)) ) ) ).

% dvd_mult_imp_div
tff(fact_1666_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D3: A,C3: A] :
          ( dvd_dvd(A,B2,A2)
         => ( dvd_dvd(A,D3,C3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),divide_divide(A,C3,D3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1667_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( ( divide_divide(A,B2,A2) = divide_divide(A,C3,A2) )
          <=> ( B2 = C3 ) ) ) ) ).

% unit_div_cancel
tff(fact_1668_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,divide_divide(A,A2,B2),C3)
          <=> dvd_dvd(A,A2,C3) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1669_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,A2,divide_divide(A,C3,B2))
          <=> dvd_dvd(A,A2,C3) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1670_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( dvd_dvd(A,C3,A2)
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1671_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,B2: A,A2: A] :
          ( dvd_dvd(A,C3,B2)
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,C3)),divide_divide(A,B2,C3)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1672_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,Nb: nat] :
          ( dvd_dvd(A,B2,A2)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,A2,B2)),Nb) = 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_1673_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
        <=> dvd_dvd(A,B2,A2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1674_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,B2)
        <=> ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1675_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
         => dvd_dvd(A,B2,A2) ) ) ).

% mod_0_imp_dvd
tff(fact_1676_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)
         => 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_1677_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat,B2: A,Mb: nat] :
          ( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
           => dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Mb),B2) ) ) ) ).

% power_le_dvd
tff(fact_1678_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A,Nb: nat,Mb: nat] :
          ( dvd_dvd(A,Xb,Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
           => 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_1679_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( modulo_modulo(A,A2,C3) = modulo_modulo(A,B2,C3) )
        <=> dvd_dvd(A,C3,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ).

% mod_eq_dvd_iff
tff(fact_1680_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : 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_1681_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)
       => ~ dvd_dvd(nat,Nb,Mb) ) ) ).

% nat_dvd_not_less
tff(fact_1682_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D2: nat,X4: nat,Y3: nat] :
          ( dvd_dvd(nat,D2,A2)
          & dvd_dvd(nat,D2,B2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1683_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( 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)
        | dvd_dvd(nat,Mb,Nb) ) ) ).

% dvd_minus_self
tff(fact_1684_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( dvd_dvd(nat,Mb,Nb)
      <=> dvd_dvd(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_1685_dvd__diffD1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( dvd_dvd(nat,K2,Mb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => dvd_dvd(nat,K2,Nb) ) ) ) ).

% dvd_diffD1
tff(fact_1686_dvd__diffD,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb))
     => ( dvd_dvd(nat,K2,Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
         => dvd_dvd(nat,K2,Mb) ) ) ) ).

% dvd_diffD
tff(fact_1687_zdvd__mono,axiom,
    ! [K2: int,Mb: int,Ta: int] :
      ( ( K2 != zero_zero(int) )
     => ( dvd_dvd(int,Mb,Ta)
      <=> dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K2),Mb),aa(int,int,aa(int,fun(int,int),times_times(int),K2),Ta)) ) ) ).

% zdvd_mono
tff(fact_1688_zdvd__mult__cancel,axiom,
    ! [K2: int,Mb: int,Nb: int] :
      ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K2),Mb),aa(int,int,aa(int,fun(int,int),times_times(int),K2),Nb))
     => ( ( K2 != zero_zero(int) )
       => dvd_dvd(int,Mb,Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_1689_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_1690_zdvd__reduce,axiom,
    ! [K2: int,Nb: int,Mb: int] :
      ( dvd_dvd(int,K2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K2),Mb)))
    <=> dvd_dvd(int,K2,Nb) ) ).

% zdvd_reduce
tff(fact_1691_zdvd__period,axiom,
    ! [A2: int,D3: int,Xb: int,Ta: int,C3: int] :
      ( dvd_dvd(int,A2,D3)
     => ( dvd_dvd(int,A2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Ta))
      <=> 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),C3),D3))),Ta)) ) ) ).

% zdvd_period
tff(fact_1692_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => aa(set(nat),$o,finite_finite(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_aw(nat,fun(nat,$o),Mb))) ) ).

% finite_divisors_nat
tff(fact_1693_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ).

% unit_dvdE
tff(fact_1694_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X3: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X3))
        <=> ? [X3: A] :
              ( dvd_dvd(A,L,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),zero_zero(A)))
              & aa(A,$o,P,X3) ) ) ) ).

% unity_coeff_ex
tff(fact_1695_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 != zero_zero(A) )
         => ( dvd_dvd(A,A2,B2)
           => ( ( divide_divide(A,B2,A2) = C3 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1696_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( dvd_dvd(A,B2,A2)
           => ( dvd_dvd(A,divide_divide(A,A2,B2),C3)
            <=> dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1697_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( dvd_dvd(A,C3,B2)
           => ( dvd_dvd(A,A2,divide_divide(A,B2,C3))
            <=> dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1698_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C3 != zero_zero(A) )
           => ( dvd_dvd(A,A2,B2)
             => ( dvd_dvd(A,C3,D3)
               => ( ( divide_divide(A,B2,A2) = divide_divide(A,D3,C3) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D3) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1699_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1700_even__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: num] : 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_1701_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D4: A,Ta: A] :
          ( dvd_dvd(A,D3,D4)
         => ! [X: A,K4: A] :
              ( ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Ta))
            <=> ~ dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),Ta)) ) ) ) ).

% inf_period(4)
tff(fact_1702_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D4: A,Ta: A] :
          ( dvd_dvd(A,D3,D4)
         => ! [X: A,K4: A] :
              ( dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Ta))
            <=> dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),Ta)) ) ) ) ).

% inf_period(3)
tff(fact_1703_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( divide_divide(A,A2,B2) = C3 )
          <=> ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_1704_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( ( A2 = divide_divide(A,C3,B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C3 ) ) ) ) ).

% unit_eq_div2
tff(fact_1705_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( dvd_dvd(A,C3,one_one(A))
         => ( dvd_dvd(A,B2,A2)
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A2,B2),C3) ) ) ) ) ).

% div_mult_unit2
tff(fact_1706_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),C3) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),B2) ) ) ) ).

% unit_div_commute
tff(fact_1707_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A2: A,B2: A] :
          ( dvd_dvd(A,C3,one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ) ) ).

% unit_div_mult_swap
tff(fact_1708_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C3: A,A2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( dvd_dvd(A,C3,one_one(A))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = divide_divide(A,divide_divide(A,A2,B2),C3) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_1709_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1710_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),one_one(A))
        <=> ( dvd_dvd(A,A2,one_one(A))
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1711_dvd__imp__le,axiom,
    ! [K2: nat,Nb: nat] :
      ( dvd_dvd(nat,K2,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),K2),Nb) ) ) ).

% dvd_imp_le
tff(fact_1712_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
      <=> dvd_dvd(nat,Mb,Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1713_dvd__mult__cancel,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => dvd_dvd(nat,Mb,Nb) ) ) ).

% dvd_mult_cancel
tff(fact_1714_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))
    <=> ~ dvd_dvd(nat,Nb,Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1715_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( ( modulo_modulo(nat,Mb,Q3) = modulo_modulo(nat,Nb,Q3) )
      <=> dvd_dvd(nat,Q3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1716_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K2: A,Mb: fun(A,nat)] :
      ( aa(A,$o,P,K2)
     => ? [X4: A] :
          ( aa(A,$o,P,X4)
          & ! [Y4: A] :
              ( aa(A,$o,P,Y4)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Mb,X4)),aa(A,nat,Mb,Y4)) ) ) ) ).

% ex_has_least_nat
tff(fact_1717_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),zero_zero(A)) ) ).

% even_zero
tff(fact_1718_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B4: A] :
                  ( ( B4 != zero_zero(A) )
                 => ( dvd_dvd(A,B4,one_one(A))
                   => ( ( divide_divide(A,one_one(A),A2) = B4 )
                     => ( ( divide_divide(A,one_one(A),B4) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B4) = one_one(A) )
                         => ( divide_divide(A,C3,A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C3),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1719_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( dvd_dvd(A,B2,one_one(A))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_1720_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( dvd_dvd(A,B2,one_one(A))
           => ( divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = divide_divide(A,one_one(A),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_1721_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ~ ! [B4: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B4) ) ) ).

% evenE
tff(fact_1722_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),one_one(A)) ) ).

% odd_one
tff(fact_1723_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)
           => 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_1724_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
            <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) )
            & ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = divide_divide(A,B2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_1725_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xb: A,Mb: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Mb),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb))
          <=> ( dvd_dvd(A,Xb,one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ) ).

% dvd_power_iff
tff(fact_1726_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) ) )
         => dvd_dvd(A,Xb,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) ) ) ).

% dvd_power
tff(fact_1727_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( divide_divide(nat,Xb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = divide_divide(nat,Y,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
     => ( ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Xb)
        <=> dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Y) )
       => ( Xb = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_1728_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
    <=> 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_1729_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( 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_1730_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Mb)
     => ( 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_1731_dvd__minus__add,axiom,
    ! [Q3: nat,Nb: nat,R2: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),Mb))
       => ( dvd_dvd(nat,Mb,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Q3))
        <=> 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),R2),Mb)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_1732_power__dvd__imp__le,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Nb))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_1733_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Mb)
       => ( dvd_dvd(nat,Nb,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),R2))
         => ( modulo_modulo(nat,Mb,Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_1734_aset_I10_J,axiom,
    ! [D3: int,D4: int,A5: set(int),Ta: int] :
      ( dvd_dvd(int,D3,D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A5)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Ta))
           => ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)),Ta)) ) ) ) ).

% aset(10)
tff(fact_1735_aset_I9_J,axiom,
    ! [D3: int,D4: int,A5: set(int),Ta: int] :
      ( dvd_dvd(int,D3,D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A5)
                 => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa3) ) ) )
         => ( dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Ta))
           => dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)),Ta)) ) ) ) ).

% aset(9)
tff(fact_1736_bset_I10_J,axiom,
    ! [D3: int,D4: int,B5: set(int),Ta: int] :
      ( dvd_dvd(int,D3,D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Ta))
           => ~ dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)),Ta)) ) ) ) ).

% bset(10)
tff(fact_1737_bset_I9_J,axiom,
    ! [D3: int,D4: int,B5: set(int),Ta: int] :
      ( dvd_dvd(int,D3,D4)
     => ! [X: int] :
          ( ! [Xa3: int] :
              ( member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa3) ) ) )
         => ( dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Ta))
           => dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)),Ta)) ) ) ) ).

% bset(9)
tff(fact_1738_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( 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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_1739_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( 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_1740_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ 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_1741_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
         => ( aa(A,$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_1742_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ 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_1743_dvd__power__iff__le,axiom,
    ! [K2: 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))),K2)
     => ( dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K2),Mb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K2),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_1744_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( 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))
        <=> ( 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_1745_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( 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))
        <=> ( 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_1746_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),bit_se8732182000553998342ip_bit(A,Mb,A2))
        <=> ~ ( 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_1747_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( 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),K2),L))
    <=> 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),K2),L)) ) ).

% even_diff_iff
tff(fact_1748_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ~ ! [B4: 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))),B4)),one_one(A)) ) ) ).

% oddE
tff(fact_1749_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(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_1750_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( 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) ) )
         => ~ ( ~ 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_1751_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_1752_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
         => ( aa(A,$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_1753_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
         => aa(A,$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_1754_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( ( Nb = zero_zero(nat) )
            | ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
              & ( A2 != zero_zero(A) ) )
            | ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_1755_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K2: A,F3: fun(A,nat),B2: nat] :
      ( aa(A,$o,P,K2)
     => ( ! [Y3: A] :
            ( aa(A,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Y3)),B2) )
       => ? [X4: A] :
            ( aa(A,$o,P,X4)
            & ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,Y4)),aa(A,nat,F3,X4)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_1756_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B4)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B4),A4) )
     => ( ! [A4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A4),zero_zero(nat))
       => ( ! [A4: nat,B4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B4)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B4)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_1757_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ).

% even_mask_div_iff'
tff(fact_1758_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,$o),ord_less(nat),zero_zero(nat)),Nb)
            & ( ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( 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_1759_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_1760_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)) )
     => dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),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_1761_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ) ) ).

% even_mask_div_iff
tff(fact_1762_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)))
        <=> ( aa(nat,$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)
              & dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),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_1763_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S3: set(A)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X4: A] :
                ( member(A,X4,S3)
                & ~ ? [Xa: A] :
                      ( member(A,Xa,S3)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X4) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_1764_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X6: set(A)] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => ? [Xa: A] :
                    ( member(A,Xa,X6)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xa) ) )
           => ~ aa(set(A),$o,finite_finite(A),X6) ) ) ) ).

% infinite_growing
tff(fact_1765_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(
                      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:= 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:= 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_1766_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),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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% flip_bit_0
tff(fact_1767_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),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_1768_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( member(nat,aa(nat,nat,suc,Nb),nat_set_decode(Xb))
    <=> member(nat,Nb,nat_set_decode(divide_divide(nat,Xb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% set_decode_Suc
tff(fact_1769_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_1770_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A2: A,B2: A,C3: A,D3: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C3 != D3 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1771_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A] : aa(A,A,artanh(A),Xb) = divide_divide(A,aa(A,A,ln_ln(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_1772_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_ax(int,int)),set_or1337092689740270186AtMost(int,Mb,Nb)) = 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_1773_intind,axiom,
    ! [A: $tType,I: nat,Nb: nat,P: fun(A,$o),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
     => ( aa(A,$o,P,Xb)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Nb,Xb)),I)) ) ) ).

% intind
tff(fact_1774_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_1775_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$false) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_1776_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = zero_zero(A) )
        <=> ~ (P) ) ) ).

% of_bool_eq_0_iff
tff(fact_1777_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_1778_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_1779_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_1780_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_1781_replicate__eq__replicate,axiom,
    ! [A: $tType,Mb: nat,Xb: A,Nb: nat,Y: A] :
      ( ( replicate(A,Mb,Xb) = replicate(A,Nb,Y) )
    <=> ( ( Mb = Nb )
        & ( ( Mb != zero_zero(nat) )
         => ( Xb = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_1782_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_1783_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P)
            | (Q) )) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_or_iff
tff(fact_1784_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_1785_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_1786_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_1787_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_1788_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_1789_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_1790_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_1791_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_1792_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_1793_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K2: num] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K2)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_1794_in__set__replicate,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Y: A] :
      ( 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_1795_Bex__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
          & aa(A,$o,P,X3) )
    <=> ( aa(A,$o,P,A2)
        & ( Nb != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_1796_Ball__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
         => aa(A,$o,P,X3) )
    <=> ( aa(A,$o,P,A2)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_1797_nth__replicate,axiom,
    ! [A: $tType,I: nat,Nb: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
     => ( aa(nat,A,nth(A,replicate(A,Nb,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_1798_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_1799_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_1800_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_1801_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K2))) = 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),K2))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_1802_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [P2: $o] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> (P2) ) ) ).

% odd_of_bool_self
tff(fact_1803_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : divide_divide(A,aa($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_1804_set__decode__0,axiom,
    ! [Xb: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(Xb))
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Xb) ) ).

% set_decode_0
tff(fact_1805_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_1806_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : divide_divide(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_1807_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_1808_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( dvd_dvd(nat,Mb,Nb)
     => ( dvd_dvd(nat,Nb,Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_1809_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o,Q3: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P2)) = aa($o,A,zero_neq_one_of_bool(A),(Q3)) )
        <=> ( (P2)
          <=> (Q3) ) ) ) ).

% of_bool_eq_iff
tff(fact_1810_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_1811_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_1812_signed__take__bit__mult,axiom,
    ! [Nb: nat,K2: 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),K2)),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),K2),L)) ).

% signed_take_bit_mult
tff(fact_1813_signed__take__bit__add,axiom,
    ! [Nb: nat,K2: 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),K2)),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),K2),L)) ).

% signed_take_bit_add
tff(fact_1814_signed__take__bit__diff,axiom,
    ! [Nb: nat,K2: 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),K2)),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),K2),L)) ).

% signed_take_bit_diff
tff(fact_1815_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_1816_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,K5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),aa(A,B,G3,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),K5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),K5)) ) ) ).

% sum_mono
tff(fact_1817_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),A5: set(B),G3: fun(C,A),B5: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A5)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G3),B5)) = 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_az(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F3),G3),B5)),A5) ) ).

% sum_product
tff(fact_1818_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),A5: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A5)),R2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ba(fun(B,A),fun(A,fun(B,A)),F3),R2)),A5) ) ).

% sum_distrib_right
tff(fact_1819_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R2: A,F3: fun(B,A),A5: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A5)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bb(A,fun(fun(B,A),fun(B,A)),R2),F3)),A5) ) ).

% sum_distrib_left
tff(fact_1820_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),H: fun(B,A),A5: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bc(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G3),A5)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),A5)) ) ).

% sum.distrib
tff(fact_1821_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A5: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bd(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A5) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A5)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G3),A5)) ) ).

% sum_subtractf
tff(fact_1822_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F3: fun(B,A),A5: set(B),R2: A] : divide_divide(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A5),R2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_be(fun(B,A),fun(A,fun(B,A)),F3),R2)),A5) ) ).

% sum_divide_distrib
tff(fact_1823_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F3: fun(B,A),A2: A,A5: set(B)] : modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_bf(fun(B,A),fun(A,fun(B,A)),F3),A2)),A5),A2) = modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A5),A2) ) ).

% mod_sum_eq
tff(fact_1824_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X4)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)) ) ) ).

% sum_nonneg
tff(fact_1825_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_1826_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F3: fun(B,A),I6: set(B),G3: fun(B,A),I: B] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),I6) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G3),I6) )
         => ( ! [I2: B] :
                ( member(B,I2,I6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,I2)),aa(B,A,G3,I2)) )
           => ( member(B,I,I6)
             => ( aa(set(B),$o,finite_finite(B),I6)
               => ( aa(B,A,F3,I) = aa(B,A,G3,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_1827_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_1828_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_1829_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_1830_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_1831_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_1832_finite__set__decode,axiom,
    ! [Nb: nat] : aa(set(nat),$o,finite_finite(nat),nat_set_decode(Nb)) ).

% finite_set_decode
tff(fact_1833_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
     => ( ! [Y3: A] :
            ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( Y3 = Xb ) )
       => ( Xs = replicate(A,Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_1834_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ( X4 = Xb ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_1835_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S2: set(A),Ta: set(B),G3: fun(B,C),I: fun(B,A),F3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite(A),S2)
         => ( aa(set(B),$o,finite_finite(B),Ta)
           => ( ! [X4: B] :
                  ( member(B,X4,Ta)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G3,X4)) )
             => ( ! [X4: A] :
                    ( member(A,X4,S2)
                   => ? [Xa: B] :
                        ( member(B,Xa,Ta)
                        & ( aa(B,A,I,Xa) = X4 )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,X4)),aa(B,C,G3,Xa)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,groups7311177749621191930dd_sum(A,C,F3),S2)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G3),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_1836_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ! [X4: A] :
                ( member(A,X4,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X4)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5) = zero_zero(B) )
            <=> ! [X3: A] :
                  ( member(A,X3,A5)
                 => ( aa(A,B,F3,X3) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_1837_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ! [X4: A] :
                ( member(A,X4,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4)) )
           => ( ? [X: A] :
                  ( member(A,X,A5)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X)),aa(A,B,G3,X)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_1838_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,zero_zero(A)),zero_zero(A))
         => ( ! [X15: A,Y15: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,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)) )
           => ( aa(set(B),$o,finite_finite(B),S3)
             => ( ! [X4: B] :
                    ( member(B,X4,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X4)),aa(B,A,G3,X4)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G3),S3)) ) ) ) ) ) ).

% sum.related
tff(fact_1839_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X4: A] :
                  ( member(A,X4,A5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X4)),aa(A,B,G3,X4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5)) ) ) ) ) ).

% sum_strict_mono
tff(fact_1840_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_1841_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_1842_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_1843_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_1844_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T3: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T2: set(B),G3: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S4)
         => ( aa(set(B),$o,finite_finite(B),T3)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4))
                   => 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)) )
               => ( ! [B4: B] :
                      ( member(B,B4,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,I,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                       => member(A,aa(B,A,I,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S4)
                         => ( aa(A,C,G3,A4) = zero_zero(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T3)
                           => ( aa(B,C,H,B4) = zero_zero(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S3)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G3,A4) ) )
                         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,G3),S3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,H),T2) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_1845_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S2: set(A),F3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(A),S2)
         => ( ! [I2: A] :
                ( member(A,I2,S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),S2) = zero_zero(B) )
             => ( member(A,I,S2)
               => ( aa(A,B,F3,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_1846_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S2: set(A),F3: fun(A,B),B5: B,I: A] :
          ( aa(set(A),$o,finite_finite(A),S2)
         => ( ! [I2: A] :
                ( member(A,I2,S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),S2) = B5 )
             => ( member(A,I,S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I)),B5) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_1847_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bg(fun(A,B),fun(A,$o),G3)))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_1848_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(A),I: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I6)
         => ( member(A,I,I6)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I6)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),I6)) ) ) ) ) ) ).

% sum_pos2
tff(fact_1849_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I6)
         => ( ( I6 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I6)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),I6)) ) ) ) ) ).

% sum_pos
tff(fact_1850_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_1851_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A5))
                   => ( aa(A,B,G3,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B5))
                     => ( aa(A,B,H,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B5) )
                  <=> ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),C5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C5) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_1852_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A5))
                   => ( aa(A,B,G3,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B5))
                     => ( aa(A,B,H,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),C5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C5) )
                   => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B5) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_1853_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,G3,X4) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),S3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),T2) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_1854_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,G3,X4) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),T2) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),S3) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_1855_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,H,X4) = zero_zero(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S3)
                   => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),S3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),T2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_1856_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,G3,X4) = zero_zero(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S3)
                   => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),T2) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_1857_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B5: set(A),A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( aa(set(A),$o,finite_finite(A),A5)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),B5)) ) ) ) ) ).

% sum.subset_diff
tff(fact_1858_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),B5)) ) ) ) ) ).

% sum_diff
tff(fact_1859_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_1860_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_1861_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_1862_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),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_1863_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_1864_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,B4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),B5)) ) ) ) ) ).

% sum_mono2
tff(fact_1865_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_1866_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))),divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),Y),Y)) ) ) ).

% ln_diff_le
tff(fact_1867_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] : aa($o,A,zero_neq_one_of_bool(A),~ 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_1868_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2)),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_1869_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_ri4674362597316999326ke_bit(A,Mb),A2))
        <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ).

% even_signed_take_bit_iff
tff(fact_1870_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B5: set(A),A5: set(A),B2: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( member(A,B2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,B2))
               => ( ! [X4: A] :
                      ( member(A,X4,B5)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X4)) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),B5)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_1871_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [A4: A] :
              ( ( divide_divide(A,A4,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A4 )
             => aa(A,$o,P,A4) )
         => ( ! [A4: A,B4: $o] :
                ( aa(A,$o,P,A4)
               => ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A4)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A4 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B4))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A4))) ) )
           => aa(A,$o,P,A2) ) ) ) ).

% bits_induct
tff(fact_1872_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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_1873_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2)),K2)
    <=> 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)),K2) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_1874_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_1875_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_1876_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( ( A2 != B2 )
            & ( C3 != D3 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% crossproduct_noteq
tff(fact_1877_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_1878_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: 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)),K2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),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_1879_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_1880_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bh(nat,fun(nat,$o),Xb)) ).

% set_decode_def
tff(fact_1881_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Mb: nat,Nb: nat] :
          divide_divide(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Mb),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($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_1882_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        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:= 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:= 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_1883_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_1884_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)) = 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_1885_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),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_1886_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))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))) ) ).

% signed_take_bit_rec
tff(fact_1887_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)),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_1888_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(
                          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:= 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:= 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_1889_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_bi(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_1890_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_1891_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A2)) = A2 ) ).

% add.inverse_inverse
tff(fact_1892_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_idempotent
tff(fact_1893_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_abs
tff(fact_1894_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_1895_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_1896_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = A2 )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_1897_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_1898_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_1899_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A2) )
        <=> ( zero_zero(A) = A2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_1900_add_Oinverse__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( aa(A,A,uminus_uminus(A),zero_zero(A)) = zero_zero(A) ) ) ).

% add.inverse_neutral
tff(fact_1901_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_1902_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_1903_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_1904_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_1905_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_1906_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_1907_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_1908_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_1909_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_1910_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_1911_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ).

% div_minus_minus
tff(fact_1912_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_1913_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_1914_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_1915_abs__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_0
tff(fact_1916_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] :
          ( dvd_dvd(A,aa(A,A,uminus_uminus(A),Xb),Y)
        <=> dvd_dvd(A,Xb,Y) ) ) ).

% minus_dvd_iff
tff(fact_1917_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] :
          ( dvd_dvd(A,Xb,aa(A,A,uminus_uminus(A),Y))
        <=> dvd_dvd(A,Xb,Y) ) ) ).

% dvd_minus_iff
tff(fact_1918_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_1919_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_1920_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_1921_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_1922_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,abs_abs(A),A2),aa(A,A,abs_abs(A),B2)) ) ).

% abs_divide
tff(fact_1923_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus_cancel
tff(fact_1924_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus
tff(fact_1925_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_1926_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: A,K2: A] :
          ( dvd_dvd(A,Mb,aa(A,A,abs_abs(A),K2))
        <=> dvd_dvd(A,Mb,K2) ) ) ).

% dvd_abs_iff
tff(fact_1927_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Mb: A,K2: A] :
          ( dvd_dvd(A,aa(A,A,abs_abs(A),Mb),K2)
        <=> dvd_dvd(A,Mb,K2) ) ) ).

% abs_dvd_iff
tff(fact_1928_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: $o] : aa(A,A,abs_abs(A),aa($o,A,zero_neq_one_of_bool(A),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% abs_bool_eq
tff(fact_1929_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: 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),F3),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),F3,A2),B2) ).

% case_prod_conv
tff(fact_1930_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_1931_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_1932_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_1933_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_1934_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_1935_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_1936_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_1937_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_1938_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_1939_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_1940_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_1941_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_1942_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_1943_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_1944_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_1945_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_1946_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_1947_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_1948_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_1949_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_1950_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_1951_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: 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_1952_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: 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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_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_1959_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A5))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_bj(fun(B,A),fun(B,A),F3)),A5)) ) ).

% sum_abs
tff(fact_1960_dbl__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K2))) ) ).

% dbl_simps(1)
tff(fact_1961_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_1962_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_1963_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_1964_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_1965_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_1966_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A2,aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$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_1967_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_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_1974_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_1975_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_1976_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_1977_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_1978_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_1979_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_1980_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_1981_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_1982_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_1983_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_1984_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_1985_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_1986_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_1987_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_bj(fun(B,A),fun(B,A),F3)),A5)) ) ).

% sum_abs_ge_zero
tff(fact_1988_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_1989_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,$o),ord_less_eq(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$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_1990_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,$o),ord_less_eq(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$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_1991_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: 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_1992_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = 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_1993_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_1994_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,$o),ord_less(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$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_1995_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,$o),ord_less(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$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_1996_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_1997_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_1998_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_1999_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_2000_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb)))) ) ).

% sum.cl_ivl_Suc
tff(fact_2001_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: fun(nat,A),A5: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bk(fun(nat,A),fun(nat,A),C3)),A5) = $ite(
            ( aa(set(nat),$o,finite_finite(nat),A5)
            & member(nat,zero_zero(nat),A5) ),
            aa(nat,A,C3,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_2002_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_2003_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_2004_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_2005_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(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_2006_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_2007_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_2008_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_2009_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A2: A] :
          ( ~ 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_2010_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A2: A] :
          ( 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_2011_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A2: A] :
          ( 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_2012_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_2013_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_2014_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: 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,K2)))) = 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),K2)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_2015_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_2016_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: fun(nat,A),D3: fun(nat,A),A5: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bl(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D3)),A5) = $ite(
            ( aa(set(nat),$o,finite_finite(nat),A5)
            & member(nat,zero_zero(nat),A5) ),
            divide_divide(A,aa(nat,A,C3,zero_zero(nat)),aa(nat,A,D3,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_2017_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_2018_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          ( ~ 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_2019_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          ( 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_2020_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_2021_signed__take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K2)) ).

% signed_take_bit_minus
tff(fact_2022_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_2023_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_2024_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_2025_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_2026_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,H: fun(B,A),F3: 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),F3),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_bm(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),H),F3)),Prod) ).

% prod.case_distrib
tff(fact_2027_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A2 ) ) ) ).

% minus_equation_iff
tff(fact_2028_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% equation_minus_iff
tff(fact_2029_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,abs_abs(A),Xb) = aa(A,A,abs_abs(A),Y) )
        <=> ( ( Xb = Y )
            | ( Xb = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_2030_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_2031_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_2032_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_2033_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_2034_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_2035_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_2036_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X: A] :
          aa(A,A,abs_abs(A),X) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),zero_zero(A)),aa(A,A,uminus_uminus(A),X),X) ) ).

% abs_if_raw
tff(fact_2037_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_2038_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_2039_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_2040_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_2041_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_2042_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2043_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_2044_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_2045_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_2046_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K2: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K2) )
         => dvd_dvd(A,L,K2) ) ) ).

% dvd_if_abs_eq
tff(fact_2047_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_2048_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_2049_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_2050_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_2051_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_2052_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_2053_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_2054_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_2055_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_2056_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_2057_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_2058_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_2059_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_2060_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_2061_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_2062_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_2063_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A5: A,K2: A,A2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),A2) )
         => ( aa(A,A,uminus_uminus(A),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K2)),aa(A,A,uminus_uminus(A),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_2064_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_2065_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_2066_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_2067_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_2068_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% div_minus_right
tff(fact_2069_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_divide_right
tff(fact_2070_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ).

% minus_divide_divide
tff(fact_2071_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% minus_divide_left
tff(fact_2072_old_Oprod_Ocase,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),X1: B,X2: 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),F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),X2)) = aa(C,A,aa(B,fun(C,A),F3,X1),X2) ).

% old.prod.case
tff(fact_2073_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_2074_mod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A3: A] :
          ( ( modulo_modulo(A,A2,B2) = modulo_modulo(A,A3,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A3),B2) ) ) ) ).

% mod_minus_cong
tff(fact_2075_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_2076_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),G3: fun(product_prod(A,B),C)] :
      ( ! [X4: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F3,X4),Y3) = aa(product_prod(A,B),C,G3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3) = G3 ) ) ).

% cond_case_prod_eta
tff(fact_2077_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: 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_bn(fun(product_prod(A,B),C),fun(A,fun(B,C)),F3)) = F3 ).

% case_prod_eta
tff(fact_2078_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,$o,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z))
     => ~ ! [X4: B,Y3: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y3) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X4),Y3)) ) ) ).

% case_prodE2
tff(fact_2079_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_2080_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_2081_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_2082_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_2083_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D3)
           => aa(A,$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),C3),D3)) ) ) ) ).

% abs_mult_less
tff(fact_2084_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_2085_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_2086_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_2087_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,abs_abs(A),A2),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_2088_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(nat),F3: fun(nat,A),G3: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A5)
         => ( ! [X4: nat] :
                ( member(nat,aa(nat,nat,suc,X4),A5)
               => ( aa(nat,A,F3,aa(nat,nat,suc,X4)) = aa(nat,A,G3,aa(nat,nat,suc,X4)) ) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),A5) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),A5) ) ) ) ) ).

% sum_cong_Suc
tff(fact_2089_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_2090_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_2091_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_2092_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_2093_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_2094_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_2095_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_2096_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_2097_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_2098_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_2099_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_2100_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_2101_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_2102_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_2103_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_2104_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_2105_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),aa(A,A,uminus_uminus(A),B2)) = divide_divide(A,A2,B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2106_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_2107_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_2108_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_2109_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_2110_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B5: A,K2: A,B2: A,A2: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_2111_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_2112_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_2113_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A2),B2) = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) ) ) ) ).

% dvd_neg_div
tff(fact_2114_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,A2)
         => ( divide_divide(A,A2,aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) ) ) ) ).

% dvd_div_neg
tff(fact_2115_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_2116_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_2117_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_2118_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_2119_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_2120_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_2121_sum__subtractf__nat,axiom,
    ! [A: $tType,A5: set(A),G3: fun(A,nat),F3: fun(A,nat)] :
      ( ! [X4: A] :
          ( member(A,X4,A5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G3,X4)),aa(A,nat,F3,X4)) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_bo(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G3),F3)),A5) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,G3),A5)) ) ) ).

% sum_subtractf_nat
tff(fact_2122_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2123_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2124_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_2125_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_2126_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_2127_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_2128_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)
         => ( divide_divide(A,aa(A,A,abs_abs(A),Xb),Y) = aa(A,A,abs_abs(A),divide_divide(A,Xb,Y)) ) ) ) ).

% abs_div_pos
tff(fact_2129_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_2130_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_le_iff
tff(fact_2131_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_2132_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C3: A,D3: A] : aa(A,$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),C3),D3)))),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),C3))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)))) ) ).

% abs_diff_triangle_ineq
tff(fact_2133_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_less_iff
tff(fact_2134_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X3: A] :
            ( member(A,X3,A5)
            & ( aa(A,nat,F3,X3) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa2: A] :
                ( member(A,Xa2,A5)
               => ( ( X3 != Xa2 )
                 => ( aa(A,nat,F3,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_2135_sum__SucD,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A),Nb: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5) = aa(nat,nat,suc,Nb) )
     => ? [X4: A] :
          ( member(A,X4,A5)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X4)) ) ) ).

% sum_SucD
tff(fact_2136_sum__eq__1__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5) = one_one(nat) )
      <=> ? [X3: A] :
            ( member(A,X3,A5)
            & ( aa(A,nat,F3,X3) = one_one(nat) )
            & ! [Xa2: A] :
                ( member(A,Xa2,A5)
               => ( ( X3 != Xa2 )
                 => ( aa(A,nat,F3,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2137_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_2138_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_2139_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_2140_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_2141_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_2142_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_2143_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_2144_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_2145_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_2146_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_2147_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_2148_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_2149_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_2150_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_2151_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_2152_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_2153_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_2154_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_2155_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = aa(A,A,uminus_uminus(A),B2),A2 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_2156_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)) = A2 )
        <=> $ite(C3 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),A2 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_2157_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) = C3 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2158_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C3 = aa(A,A,uminus_uminus(A),divide_divide(A,A2,B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2159_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_2160_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_2161_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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_2162_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_2163_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_2164_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K2: 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,K2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K2))) ) ).

% power_minus_Bit0
tff(fact_2165_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)
       => ? [D2: 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(real),A2),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2166_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_2167_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_2168_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_2169_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_2170_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Mb: nat,I6: set(nat)] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_br(A,fun(nat,fun(nat,A)),Xb),Mb)),I6) = 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)),I6)) ) ).

% sum_power_add
tff(fact_2171_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_2172_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_2173_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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_bs(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2174_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_bt(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_bu(nat,fun(complex,$o),Nb))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_2175_sum__nth__roots,axiom,
    ! [Nb: nat,C3: 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_bt(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_aq(nat,fun(complex,fun(complex,$o)),Nb),C3))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_2176_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_2177_sum__diff__nat,axiom,
    ! [A: $tType,B5: set(A),A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),B5)) ) ) ) ).

% sum_diff_nat
tff(fact_2178_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,zero_zero(nat),K2)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_2179_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2180_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,aa(nat,nat,suc,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2181_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2182_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
          <=> aa(A,$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),C3)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2183_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2184_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2185_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
          <=> aa(A,$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),C3)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2186_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_2187_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_2188_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,W: num] :
          ( ( divide_divide(A,B2,C3) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(C3 != 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))),C3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2189_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = divide_divide(A,B2,C3) )
        <=> $ite(C3 != 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))),C3) = B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2190_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),divide_divide(A,Xb,Z))),Y) = 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_2191_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),divide_divide(A,A2,Z))),B2) = $ite(Z = zero_zero(A),B2,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_2192_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),divide_divide(A,Xb,Z))),Y) = 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_2193_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),divide_divide(A,A2,Z)),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),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_2194_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),divide_divide(A,A2,Z))),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),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_2195_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,uminus_uminus(A),A2))
        <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ).

% even_minus
tff(fact_2196_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)
       => ? [D2: 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),A2),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_2197_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_2198_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_2199_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$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,G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2200_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F3: fun(nat,A)] :
          ( aa(nat,$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_bv(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,aa(nat,nat,suc,Nb))),aa(nat,A,F3,Mb)) ) ) ) ).

% sum_Suc_diff
tff(fact_2201_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_2202_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_2203_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_2204_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( 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_2205_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A),P2: nat] :
          ( aa(nat,$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,G3),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,G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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_2206_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
          <=> aa(A,$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),C3)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2207_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2208_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2209_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
          <=> aa(A,$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),C3)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2210_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_2211_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C3)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_2212_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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))),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3)),aa(A,$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_2213_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B2,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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))),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3)),B2),aa(A,$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_2214_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_2215_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(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_2216_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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),K2)) = 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),K2)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2217_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_2218_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_2219_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K2: 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),K2)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2220_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2)),K2)
    <=> 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))),K2) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2221_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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_2222_minus__mod__int__eq,axiom,
    ! [L: int,K2: 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),K2),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),K2),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2223_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_2224_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(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),divide_divide(int,A2,B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2225_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(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),divide_divide(int,A2,B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2226_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),Xb: A] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X4)
             => aa(A,$o,aa(A,fun(A,$o),P,X4),aa(nat,A,aa(A,fun(nat,A),power_power(A),X4),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_2227_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_2228_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_2229_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_2230_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
         => ( aa(A,$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_2231_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R2 = zero_zero(int),aa(int,int,uminus_uminus(int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q3)),one_one(int)))),
              $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_2232_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B2,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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))),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3)),B2),aa(A,$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_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),divide_divide(A,B2,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$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))),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C3)),aa(A,$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_2234_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),Xb: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( member(A,I2,I6)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xb,I2)) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xb),I6) = one_one(B) )
           => ( ! [I2: A] :
                  ( member(A,I2,I6)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,I2)),B2))),Delta) )
             => aa(B,$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_bw(fun(A,B),fun(fun(A,B),fun(A,B)),Xb),A2)),I6)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2235_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),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_2236_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_2237_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_2238_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(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_2239_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bx(fun(nat,A),fun(nat,A),F3)),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,F3,Mb)),aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_2240_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_by(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,Mb)) ) ) ) ).

% sum_telescope''
tff(fact_2241_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2) = K2 )
    <=> ( 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))),K2)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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_2242_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K2: 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))),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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),K2) = K2 ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2243_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : 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_2244_div__pos__neg__trivial,axiom,
    ! [K2: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),L)),zero_zero(int))
       => ( divide_divide(int,K2,L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_2245_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,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Nb))) ) ).

% mask_eq_sum_exp
tff(fact_2246_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_2247_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_2248_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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_bz(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.in_pairs
tff(fact_2249_int__bit__induct,axiom,
    ! [P: fun(int,$o),K2: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K: int] :
              ( aa(int,$o,P,K)
             => ( ( K != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) ) )
         => ( ! [K: int] :
                ( aa(int,$o,P,K)
               => ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) ) )
           => aa(int,$o,P,K2) ) ) ) ) ).

% int_bit_induct
tff(fact_2250_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_2251_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_ai(nat,fun(nat,$o)),Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_2252_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ca(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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_2253_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_cb(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_2254_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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),K2),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),K2)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2255_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_cc(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_2256_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_2257_arith__series__nat,axiom,
    ! [A2: nat,D3: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_cd(nat,fun(nat,fun(nat,nat)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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),D3))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% arith_series_nat
tff(fact_2258_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ca(nat,nat)),set_or1337092689740270186AtMost(nat,Mb,Nb)) = 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_2259_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,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_2260_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_ce(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Mb,Nb)) ) ).

% divmod_algorithm_code(5)
tff(fact_2261_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_cf(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_2262_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: 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,K2)))) = 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),K2))),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_2263_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)),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_2264_even__set__encode__iff,axiom,
    ! [A5: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),A5)
     => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(set(nat),nat,nat_set_encode,A5))
      <=> ~ member(nat,zero_zero(nat),A5) ) ) ).

% even_set_encode_iff
tff(fact_2265_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K2: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K2))) = 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),K2))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_2266_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_2267_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_2268_case__prodI,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F3,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),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) ) ).

% case_prodI
tff(fact_2269_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C3: fun(A,fun(B,$o))] :
      ( ! [A4: A,B4: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => aa(B,$o,aa(A,fun(B,$o),C3,A4),B4) )
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C3),P2) ) ).

% case_prodI2
tff(fact_2270_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C3: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C3,A2),B2))
     => 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)),C3),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_2271_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C3: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B4: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => member(C,Z,aa(B,set(C),aa(A,fun(B,set(C)),C3,A4),B4)) )
     => 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)),C3),P2)) ) ).

% mem_case_prodI2
tff(fact_2272_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C3: fun(A,fun(B,fun(C,$o))),Xb: C] :
      ( ! [A4: A,B4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) = P2 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C3,A4),B4),Xb) )
     => aa(C,$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)),C3),P2),Xb) ) ).

% case_prodI2'
tff(fact_2273_semiring__norm_I88_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,bit0,Mb) != aa(num,num,bit1,Nb) ).

% semiring_norm(88)
tff(fact_2274_semiring__norm_I89_J,axiom,
    ! [Mb: num,Nb: num] : aa(num,num,bit1,Mb) != aa(num,num,bit0,Nb) ).

% semiring_norm(89)
tff(fact_2275_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one2 != aa(num,num,bit1,Nb) ).

% semiring_norm(84)
tff(fact_2276_semiring__norm_I86_J,axiom,
    ! [Mb: num] : aa(num,num,bit1,Mb) != one2 ).

% semiring_norm(86)
tff(fact_2277_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_2278_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_2279_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( dvd_dvd(int,Xb,one_one(int))
    <=> ( aa(int,int,abs_abs(int),Xb) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_2280_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_2281_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_2282_set__decode__inverse,axiom,
    ! [Nb: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(Nb)) = Nb ).

% set_decode_inverse
tff(fact_2283_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_2284_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_2285_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_2286_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_2287_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_2288_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_2289_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_2290_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_2291_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_2292_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_2293_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_2294_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_2295_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_2296_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_2297_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_2298_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_2299_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_2300_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_2301_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_2302_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_2303_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_2304_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_2305_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_2306_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_2307_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_2308_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_2309_set__encode__inverse,axiom,
    ! [A5: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),A5)
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A5)) = A5 ) ) ).

% set_encode_inverse
tff(fact_2310_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_2311_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_2312_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_2313_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_2314_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_2315_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : 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))) = divide_divide(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_2316_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_2317_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_2318_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_2319_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_2320_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_2321_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_2322_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_2323_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_2324_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_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Nb: num] :
          ( 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_2332_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_2333_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_2334_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V: num] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),aa(num,nat,numeral_numeral(nat),V)) = 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_2335_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : divide_divide(nat,Mb,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))) = 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_2336_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_2337_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_2338_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_2339_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_2340_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb) ) ) ).

% even_of_nat
tff(fact_2341_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$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),I)),Nb)),Xb) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2342_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb))
        <=> aa(nat,$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),I)),Nb)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2343_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb))
        <=> aa(nat,$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),I)),Nb)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2344_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$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),I)),Nb)),Xb) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2345_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_2346_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_2347_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_2348_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_cg(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,Mb,Nb)) ) ).

% divmod_algorithm_code(6)
tff(fact_2349_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_2350_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z: A,C3: fun(B,fun(C,set(A))),P2: product_prod(B,C)] :
      ( 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)),C3),P2))
     => ~ ! [X4: B,Y3: C] :
            ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y3) )
           => ~ member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C3,X4),Y3)) ) ) ).

% mem_case_prodE
tff(fact_2351_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] : aa(num,num,bit0,X2) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_2352_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_2353_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_2354_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_2355_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_2356_case__prodD,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,$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),F3),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),F3,A2),B2) ) ).

% case_prodD
tff(fact_2357_case__prodE,axiom,
    ! [A: $tType,B: $tType,C3: fun(A,fun(B,$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),C3),P2)
     => ~ ! [X4: A,Y3: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C3,X4),Y3) ) ) ).

% case_prodE
tff(fact_2358_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,$o))),A2: A,B2: B,C3: 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,$o)),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C3)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R,A2),B2),C3) ) ).

% case_prodD'
tff(fact_2359_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(A,fun(B,fun(C,$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,$o)),C3),P2),Z)
     => ~ ! [X4: A,Y3: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C3,X4),Y3),Z) ) ) ).

% case_prodE'
tff(fact_2360_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_2361_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_2362_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_2363_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,Mb: nat,Nb: nat] : 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))) = divide_divide(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_2364_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_2365_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_2366_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_2367_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_2368_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X22: num] : Y != aa(num,num,bit0,X22)
       => ~ ! [X33: num] : Y != aa(num,num,bit1,X33) ) ) ).

% num.exhaust
tff(fact_2369_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),divide_divide(nat,Mb,Nb)) = 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_2370_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_2371_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Mb: nat,Nb: nat] :
          ( dvd_dvd(A,aa(nat,A,semiring_1_of_nat(A),Mb),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> dvd_dvd(nat,Mb,Nb) ) ) ).

% of_nat_dvd_iff
tff(fact_2372_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_2373_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_2374_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_2375_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_2376_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_2377_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_2378_set__encode__eq,axiom,
    ! [A5: set(nat),B5: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),A5)
     => ( aa(set(nat),$o,finite_finite(nat),B5)
       => ( ( aa(set(nat),nat,nat_set_encode,A5) = aa(set(nat),nat,nat_set_encode,B5) )
        <=> ( A5 = B5 ) ) ) ) ).

% set_encode_eq
tff(fact_2379_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_2380_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_2381_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_2382_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_2383_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_2384_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,A2,B2)) = 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_2385_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_2386_nat__less__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_less_as_int
tff(fact_2387_nat__leq__as__int,axiom,
    ! [X: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_leq_as_int
tff(fact_2388_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_2389_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_2390_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_2391_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q3: 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,Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) ) ).

% cong_exp_iff_simps(10)
tff(fact_2392_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q3: 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,Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) ) ).

% cong_exp_iff_simps(12)
tff(fact_2393_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q3: 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,Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q3)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q3)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_2394_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K2: 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,K2))) = 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,K2)))) ) ).

% power_minus_Bit1
tff(fact_2395_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero(int) )
     => ( ! [N: nat] :
            ( ( K2 = 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] :
              ( ( K2 = 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_2396_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ! [Y4: real] :
        ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),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_2397_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_2398_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_2399_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_2400_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),divide_divide(nat,Nb,Xb))),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_2401_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_2402_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_2403_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_2404_abs__mod__less,axiom,
    ! [L: int,K2: 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,K2,L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_2405_real__of__nat__div,axiom,
    ! [D3: nat,Nb: nat] :
      ( dvd_dvd(nat,D3,Nb)
     => ( aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Nb,D3)) = divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Nb),aa(nat,real,semiring_1_of_nat(real),D3)) ) ) ).

% real_of_nat_div
tff(fact_2406_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_2407_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_2408_set__encode__inf,axiom,
    ! [A5: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite(nat),A5)
     => ( aa(set(nat),nat,nat_set_encode,A5) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_2409_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,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_2410_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : 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_2411_odd__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: num] : ~ 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_2412_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q3: 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,Q3))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2413_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_2414_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_2415_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_2416_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),divide_divide(nat,Mb,Nb)) = 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_2417_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
          & ( K2 = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2418_pos__int__cases,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ~ ! [N: nat] :
            ( ( K2 = 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_2419_neg__int__cases,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
     => ~ ! [N: nat] :
            ( ( K2 = 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_2420_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_2421_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_2422_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K2: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => 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),K2)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K2)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_2423_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero(int) )
     => ( 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_2424_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_2425_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D3: nat] : divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Xb),aa(nat,real,semiring_1_of_nat(real),D3)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Xb,D3))),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),modulo_modulo(nat,Xb,D3)),aa(nat,real,semiring_1_of_nat(real),D3))) ).

% real_of_nat_div_aux
tff(fact_2426_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_2427_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_2428_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),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Mb))),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ) ) ).

% inverse_of_nat_le
tff(fact_2429_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: num,Nb: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Nb),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2430_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Mb: num,Q3: 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,Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2431_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( 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),K2)),L))
    <=> 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),K2),L)) ) ).

% even_abs_add_iff
tff(fact_2432_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( 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),K2),aa(int,int,abs_abs(int),L)))
    <=> 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),K2),L)) ) ).

% even_add_abs_iff
tff(fact_2433_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C3: 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)),C3)
       => ( ! [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)),C3) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2434_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Mb))),Nb) = 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_2435_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_2436_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_2437_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),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),divide_divide(nat,Nb,Xb)))) ).

% real_of_nat_div2
tff(fact_2438_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),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),divide_divide(nat,Nb,Xb)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_2439_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_2440_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),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_2441_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),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_2442_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F3: fun(nat,int),K2: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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,F3,aa(nat,nat,suc,I2))),aa(nat,int,F3,I2)))),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,F3,Mb)),K2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(nat,int,F3,Nb))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
                & ( aa(nat,int,F3,I2) = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2443_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_2444_incr__lemma,axiom,
    ! [D3: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => 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))),D3))) ) ).

% incr_lemma
tff(fact_2445_decr__lemma,axiom,
    ! [D3: int,Xb: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => 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))),D3))),Z) ) ).

% decr_lemma
tff(fact_2446_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_2447_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_2448_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_2449_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),divide_divide(nat,Mb,Nb)),modulo_modulo(nat,Mb,Nb)) ).

% divmod_nat_def
tff(fact_2450_nat__ivt__aux,axiom,
    ! [Nb: nat,F3: fun(nat,int),K2: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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,F3,aa(nat,nat,suc,I2))),aa(nat,int,F3,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(nat,int,F3,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F3,I2) = K2 ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2451_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F3: fun(nat,int),K2: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),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,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F3,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(nat,int,F3,Nb))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,int,F3,I2) = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2452_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_2453_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D3: 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_ch(A,fun(A,fun(nat,A)),A2),D3)),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)),D3))) ) ).

% double_arith_series
tff(fact_2454_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,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_2455_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)) = 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_2456_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D3: A,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ci(A,fun(A,fun(nat,A)),A2),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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)),D3))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% arith_series
tff(fact_2457_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_2458_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( 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_2459_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)),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_2460_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_2461_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)) )
     => ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),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_2462_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)) = 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_2463_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_cj(nat,fun(nat,A))),divmod_nat(Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% of_nat_code_if
tff(fact_2464_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E3)
         => ~ ! [N: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),E3) ) ) ).

% nat_approx_posE
tff(fact_2465_one__div__minus__numeral,axiom,
    ! [Nb: num] : 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_2466_minus__one__div__numeral,axiom,
    ! [Nb: num] : 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_2467_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_ck(real,fun(nat,real),Xb)) ) ).

% monoseq_arctan_series
tff(fact_2468_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),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_2469_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X: 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_cl($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X)
    <=> ( (P)
        & aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q),X) ) ) ).

% split_part
tff(fact_2470_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,fun(B,$o)),B5: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A5),B5)
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),$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,$o),A5))),aa(fun(product_prod(A,B),$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),B5))) ) ).

% Collect_case_prod_mono
tff(fact_2471_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,$o),aTP_Lamp_cm(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_2472_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_2473_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),K6) )
        <=> ? [N5: nat] :
            ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N5))) ) ) ).

% lemma_NBseq_def
tff(fact_2474_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K6)
              & ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),K6) )
        <=> ? [N5: nat] :
            ! [N4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N5))) ) ) ).

% lemma_NBseq_def2
tff(fact_2475_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_2476_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_2477_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_2478_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_1: nat] : aa(nat,$o,P,X_1)
       => ? [N: nat] :
            ( ~ aa(nat,$o,P,N)
            & aa(nat,$o,P,aa(nat,nat,suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_2479_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_2480_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),W))) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_2481_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_2482_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_2483_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_2484_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_2485_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_2486_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_2487_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_2488_norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,divide_divide(A,A2,B2)) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),real_V7770717601297561774m_norm(A,B2)) ) ).

% norm_divide
tff(fact_2489_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_2490_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_2491_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,divide_divide(A,A2,B2)) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2492_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_2493_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A,R2: real,Y: A,S2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S2)
           => aa(real,$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),R2),S2)) ) ) ) ).

% norm_mult_less
tff(fact_2494_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_2495_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E3)
         => aa(real,$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))),E3) ) ) ).

% norm_triangle_lt
tff(fact_2496_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,R2: real,Y: A,S2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S2)
           => aa(real,$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),R2),S2)) ) ) ) ).

% norm_add_less
tff(fact_2497_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_2498_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R2: real,B2: A,S2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),S2)
           => aa(real,$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),R2),S2)) ) ) ) ).

% norm_triangle_mono
tff(fact_2499_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_2500_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E3)
         => aa(real,$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))),E3) ) ) ).

% norm_triangle_le
tff(fact_2501_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C3)
         => aa(real,$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)),C3)) ) ) ).

% norm_add_leD
tff(fact_2502_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_2503_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_2504_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_2505_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_2506_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E3)
         => aa(real,$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))),E3) ) ) ).

% norm_triangle_le_diff
tff(fact_2507_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_2508_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_2509_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_2510_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C3: A,D3: A] : aa(real,$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),C3),D3)))),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),C3))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2511_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_2512_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_2513_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_2514_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: 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,K2)))) = 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),K2))),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_2515_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_cn(real,fun(nat,real),Xb)) ) ) ) ).

% ln_series
tff(fact_2516_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_2517_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_co(real,fun(nat,real),Xb)) ) ) ).

% arctan_series
tff(fact_2518_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),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_cq(A,fun(A,fun(nat,fun(nat,A))),H),Z),Nb)),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_2519_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_co(real,fun(nat,real),Xb)) ) ).

% summable_arctan_series
tff(fact_2520_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K2: A] :
          ( member(A,I,set_ord_lessThan(A,K2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),K2) ) ) ).

% lessThan_iff
tff(fact_2521_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_2522_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_lessThan(A,Xb)),set_ord_lessThan(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% lessThan_subset_iff
tff(fact_2523_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2524_eq__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K2) = aa(nat,nat,suc,Nb) )
    <=> ( pred_numeral(K2) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_2525_Suc__eq__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( ( aa(nat,nat,suc,Nb) = aa(num,nat,numeral_numeral(nat),K2) )
    <=> ( Nb = pred_numeral(K2) ) ) ).

% Suc_eq_numeral
tff(fact_2526_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),set_ord_lessThan(nat,Nb))),aa(nat,A,G3,Nb)) ) ).

% sum.lessThan_Suc
tff(fact_2527_less__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K2)),Nb) ) ).

% less_numeral_Suc
tff(fact_2528_less__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),pred_numeral(K2)) ) ).

% less_Suc_numeral
tff(fact_2529_pred__numeral__simps_I3_J,axiom,
    ! [K2: num] : pred_numeral(aa(num,num,bit1,K2)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K2)) ).

% pred_numeral_simps(3)
tff(fact_2530_le__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),pred_numeral(K2)) ) ).

% le_Suc_numeral
tff(fact_2531_le__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K2)),Nb) ) ).

% le_numeral_Suc
tff(fact_2532_diff__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K2)),Nb) ).

% diff_numeral_Suc
tff(fact_2533_diff__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),pred_numeral(K2)) ).

% diff_Suc_numeral
tff(fact_2534_max__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K2)),Nb)) ).

% max_numeral_Suc
tff(fact_2535_max__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K2)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Nb),pred_numeral(K2))) ).

% max_Suc_numeral
tff(fact_2536_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_2537_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F3: fun(nat,A)] : suminf(A,aTP_Lamp_cr(fun(nat,A),fun(nat,A),F3)) = aa(nat,A,F3,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2538_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K2: 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,K2))) = 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),K2))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_2539_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: 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,K2)))) = 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),K2)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_2540_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: 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,K2))) = 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),K2))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_2541_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cs(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_2542_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ct(fun(nat,A),fun(A,fun(nat,A)),F3),Xb))
         => ( aa(real,$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_cu(fun(nat,A),fun(A,fun(nat,real)),F3),Z)) ) ) ) ).

% powser_insidea
tff(fact_2543_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Mb: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_ord_lessThan(A,Mb)),set_ord_lessThan(A,Nb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb) ) ) ).

% lessThan_strict_subset_iff
tff(fact_2544_numeral__eq__Suc,axiom,
    ! [K2: num] : aa(num,nat,numeral_numeral(nat),K2) = aa(nat,nat,suc,pred_numeral(K2)) ).

% numeral_eq_Suc
tff(fact_2545_sum__pos__lt__pair,axiom,
    ! [F3: fun(nat,real),K2: nat] :
      ( summable(real,F3)
     => ( ! [D2: 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,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)))),aa(nat,real,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D2)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,F3),set_ord_lessThan(nat,K2))),suminf(real,F3)) ) ) ).

% sum_pos_lt_pair
tff(fact_2546_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(nat,fun(nat,A)),G3),Nb)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_ord_lessThan(nat,Nb)) ) ).

% sum.nat_diff_reindex
tff(fact_2547_pred__numeral__def,axiom,
    ! [K2: num] : pred_numeral(K2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K2)),one_one(nat)) ).

% pred_numeral_def
tff(fact_2548_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Nb: A] :
          ( ! [X4: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X4)),aa(A,nat,P,X4))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,P),set_ord_lessThan(A,Nb))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Q),set_ord_lessThan(A,Nb))) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_cw(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,Nb)) ) ) ) ).

% sum_diff_distrib
tff(fact_2549_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),F3),Xb))
         => ( aa(real,$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_cx(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ) ).

% powser_inside
tff(fact_2550_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Nb))) ) ).

% sum.lessThan_Suc_shift
tff(fact_2551_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F3: fun(nat,A),Nb: nat,R2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),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)),R2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_cy(fun(nat,A),fun(A,fun(nat,A)),F3),R2)),set_ord_lessThan(nat,Nb)) ) ).

% sumr_diff_mult_const2
tff(fact_2552_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),F3)),set_ord_lessThan(nat,Mb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,Mb)) ) ).

% sum_lessThan_telescope'
tff(fact_2553_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bv(fun(nat,A),fun(nat,A),F3)),set_ord_lessThan(nat,Mb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,Mb)),aa(nat,A,F3,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_2554_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Nb)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_2555_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)),set_ord_lessThan(nat,Nb))) ) ).

% power_diff_1_eq
tff(fact_2556_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)),set_ord_lessThan(nat,Nb))) ) ).

% one_diff_power_eq
tff(fact_2557_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)),set_ord_lessThan(nat,Nb)) = 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_2558_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_2559_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)),set_ord_lessThan(nat,Nb)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb),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_2560_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_da(A,fun(A,fun(nat,fun(nat,A))),Z),H),Mb)),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_db(A,fun(A,fun(nat,fun(nat,A))),Z),H),Mb)),set_ord_lessThan(nat,Mb)) ) ).

% lemma_termdiff1
tff(fact_2561_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_dc(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),set_ord_lessThan(nat,Nb))) ) ).

% power_diff_sumr2
tff(fact_2562_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_dd(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,Nb)))) ) ).

% diff_power_eq_sum
tff(fact_2563_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,F3: fun(nat,A),K5: A,K2: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,P4)),K5) )
         => ( aa(A,$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,F3),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)))),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_2564_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_de(A,fun(nat,fun(nat,A)),Xb),Nb)),set_ord_lessThan(nat,Nb))) ) ).

% one_diff_power_eq'
tff(fact_2565_sum__split__even__odd,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,real),Nb: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_df(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F3),G3)),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_dg(fun(nat,real),fun(nat,real),F3)),set_ord_lessThan(nat,Nb))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_dh(fun(nat,real),fun(nat,real),G3)),set_ord_lessThan(nat,Nb))) ).

% sum_split_even_odd
tff(fact_2566_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( summable(A,aa(A,fun(nat,A),power_power(A),C3))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_2567_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_di(fun(nat,A),fun(A,fun(nat,A)),F3),C3))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F3) ) ) ) ).

% summable_divide_iff
tff(fact_2568_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => ( suminf(A,aa(A,fun(nat,A),power_power(A),C3)) = divide_divide(A,one_one(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C3)) ) ) ) ).

% suminf_geometric
tff(fact_2569_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dj(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F3) ) ) ) ).

% summable_cmult_iff
tff(fact_2570_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Nb: nat,I: nat] :
          ( summable(A,F3)
         => ( ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,M)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),I)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,I))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_ord_lessThan(nat,Nb))),suminf(A,F3)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_2571_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C3: real,N3: nat,F3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,aa(nat,nat,suc,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),C3),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N)))) )
           => summable(A,F3) ) ) ) ).

% summable_ratio_test
tff(fact_2572_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(nat,fun(nat,A)),F3),K2))
        <=> summable(A,F3) ) ) ).

% summable_iff_shift
tff(fact_2573_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_dl(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) ) ) ).

% summable_mult
tff(fact_2574_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_dm(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) ) ) ).

% summable_mult2
tff(fact_2575_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( summable(A,F3)
         => ( summable(A,G3)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3)) ) ) ) ).

% summable_add
tff(fact_2576_summable__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( summable(A,F3)
         => ( summable(A,G3)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3)) ) ) ) ).

% summable_diff
tff(fact_2577_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_di(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) ) ) ).

% summable_divide
tff(fact_2578_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,aTP_Lamp_dp(fun(nat,A),fun(nat,A),F3))
        <=> summable(A,F3) ) ) ).

% summable_Suc_iff
tff(fact_2579_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( summable(A,F3)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(nat,fun(nat,A)),F3),K2)) ) ) ).

% summable_ignore_initial_segment
tff(fact_2580_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,G3,N))
         => ( summable(A,F3)
           => ( summable(A,G3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F3)),suminf(A,G3)) ) ) ) ) ).

% suminf_le
tff(fact_2581_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dj(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
         => ( ( C3 != zero_zero(A) )
           => summable(A,F3) ) ) ) ).

% summable_mult_D
tff(fact_2582_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_2583_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_dl(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),suminf(A,F3)) ) ) ) ).

% suminf_mult
tff(fact_2584_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F3)),C3) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_dm(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) ) ) ) ).

% suminf_mult2
tff(fact_2585_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( summable(A,F3)
         => ( summable(A,G3)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,F3)),suminf(A,G3)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3)) ) ) ) ) ).

% suminf_add
tff(fact_2586_suminf__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( summable(A,F3)
         => ( summable(A,G3)
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F3)),suminf(A,G3)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3)) ) ) ) ) ).

% suminf_diff
tff(fact_2587_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,F3)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_di(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) = divide_divide(A,suminf(A,F3),C3) ) ) ) ).

% suminf_divide
tff(fact_2588_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F3)) ) ) ) ).

% suminf_nonneg
tff(fact_2589_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
           => ( ( suminf(A,F3) = zero_zero(A) )
            <=> ! [N4: nat] : aa(nat,A,F3,N4) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_2590_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F3)) ) ) ) ).

% suminf_pos
tff(fact_2591_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_dq(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_zero_power'
tff(fact_2592_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_dr(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_0_powser
tff(fact_2593_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ds(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ).

% powser_split_head(3)
tff(fact_2594_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_dt(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_ct(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ).

% summable_powser_split_head
tff(fact_2595_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),Mb: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_du(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F3),Mb),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_ct(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_2596_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( ? [N6: nat] :
            ! [N: nat] :
              ( aa(nat,$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,F3,N))),aa(nat,real,G3,N)) )
         => ( summable(real,G3)
           => summable(real,aTP_Lamp_dv(fun(nat,A),fun(nat,real),F3)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_2597_summable__rabs__comparison__test,axiom,
    ! [F3: fun(nat,real),G3: 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,F3,N))),aa(nat,real,G3,N)) )
     => ( summable(real,G3)
       => summable(real,aTP_Lamp_dw(fun(nat,real),fun(nat,real),F3)) ) ) ).

% summable_rabs_comparison_test
tff(fact_2598_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I: nat] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,I))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F3)) ) ) ) ) ).

% suminf_pos2
tff(fact_2599_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F3))
            <=> ? [I3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,I3)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_2600_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Xb: A] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_ord_lessThan(nat,N))),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F3)),Xb) ) ) ) ).

% suminf_le_const
tff(fact_2601_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G3: fun(nat,real),N3: nat,F3: fun(nat,A)] :
          ( summable(real,G3)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N))),aa(nat,real,G3,N)) )
           => summable(A,F3) ) ) ) ).

% summable_comparison_test'
tff(fact_2602_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( ? [N6: nat] :
            ! [N: nat] :
              ( aa(nat,$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,F3,N))),aa(nat,real,G3,N)) )
         => ( summable(real,G3)
           => summable(A,F3) ) ) ) ).

% summable_comparison_test
tff(fact_2603_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Xb: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_ord_lessThan(nat,N))),Xb)
           => summable(A,F3) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_2604_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),C3)) ) ) ).

% summable_geometric
tff(fact_2605_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_2606_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( suminf(A,aTP_Lamp_dp(fun(nat,A),fun(nat,A),F3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F3)),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_2607_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I6: set(nat)] :
          ( summable(A,F3)
         => ( aa(set(nat),$o,finite_finite(nat),I6)
           => ( ! [N: nat] :
                  ( member(nat,N,aa(set(nat),set(nat),uminus_uminus(set(nat)),I6))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),I6)),suminf(A,F3)) ) ) ) ) ).

% sum_le_suminf
tff(fact_2608_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( summable(A,F3)
         => ( suminf(A,F3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(nat,fun(nat,A)),F3),K2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_ord_lessThan(nat,K2))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_2609_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( summable(A,F3)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(nat,fun(nat,A)),F3),K2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F3)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_ord_lessThan(nat,K2))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_2610_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Nb: nat] :
          ( summable(A,F3)
         => ( ! [M: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,M)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_ord_lessThan(nat,Nb))),suminf(A,F3)) ) ) ) ).

% sum_less_suminf
tff(fact_2611_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F3,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ds(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_2612_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ds(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),Z) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_2613_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),E3: real] :
          ( summable(A,F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => ~ ! [N7: nat] :
                  ~ ! [M2: nat] :
                      ( aa(nat,$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,F3),set_or1337092689740270186AtMost(nat,M2,N8)))),E3) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_2614_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( summable(A,F3)
           => ? [N7: nat] :
              ! [N8: nat] :
                ( aa(nat,$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_dk(fun(nat,A),fun(nat,fun(nat,A)),F3),N8)))),R2) ) ) ) ) ).

% suminf_exist_split
tff(fact_2615_summable__power__series,axiom,
    ! [F3: fun(nat,real),Z: real] :
      ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,I2)),one_one(real))
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,I2))
       => ( 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_dx(fun(nat,real),fun(real,fun(nat,real)),F3),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_2616_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A2: fun(nat,A),M6: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),R0)
           => ( ! [N: nat] : aa(real,$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_dy(real,fun(fun(nat,A),fun(nat,real)),R2),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_2617_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mm: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),F3)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_2618_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_dz(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_2619_pi__series,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = suminf(real,aTP_Lamp_ea(nat,real)) ).

% pi_series
tff(fact_2620_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$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_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),S3))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),S3) ) ).

% pred_subset_eq2
tff(fact_2621_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_2622_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))
         => sums(A,aTP_Lamp_ec(A,fun(nat,A),Z),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_2623_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_2624_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_2625_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_2626_dbl__inc__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K2)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,K2)) ) ).

% dbl_inc_simps(5)
tff(fact_2627_dbl__dec__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K2))) ) ).

% dbl_dec_simps(1)
tff(fact_2628_dbl__inc__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K2))) ) ).

% dbl_inc_simps(1)
tff(fact_2629_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xb: A] :
          ( sums(A,aTP_Lamp_dr(fun(nat,A),fun(nat,A),A2),Xb)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xb ) ) ) ).

% powser_sums_zero_iff
tff(fact_2630_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A),S2: A,Ta: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,G3,N))
         => ( sums(A,F3,S2)
           => ( sums(A,G3,Ta)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S2),Ta) ) ) ) ) ).

% sums_le
tff(fact_2631_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),A2: A,C3: A] :
          ( sums(A,F3,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_dm(fun(nat,A),fun(A,fun(nat,A)),F3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ).

% sums_mult2
tff(fact_2632_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F3: fun(nat,A),A2: A,C3: A] :
          ( sums(A,F3,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_dl(fun(nat,A),fun(A,fun(nat,A)),F3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)) ) ) ).

% sums_mult
tff(fact_2633_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),A2: A,G3: fun(nat,A),B2: A] :
          ( sums(A,F3,A2)
         => ( sums(A,G3,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% sums_add
tff(fact_2634_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),A2: A,G3: fun(nat,A),B2: A] :
          ( sums(A,F3,A2)
         => ( sums(A,G3,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) ) ) ) ).

% sums_diff
tff(fact_2635_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),A2: A,C3: A] :
          ( sums(A,F3,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_di(fun(nat,A),fun(A,fun(nat,A)),F3),C3),divide_divide(A,A2,C3)) ) ) ).

% sums_divide
tff(fact_2636_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F3: fun(nat,A),D3: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ed(A,fun(fun(nat,A),fun(nat,A)),C3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),D3),C3))
          <=> sums(A,F3,D3) ) ) ) ).

% sums_mult2_iff
tff(fact_2637_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F3: fun(nat,A),D3: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ee(A,fun(fun(nat,A),fun(nat,A)),C3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D3))
          <=> sums(A,F3,D3) ) ) ) ).

% sums_mult_iff
tff(fact_2638_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A),A2: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dj(A,fun(fun(nat,A),fun(nat,A)),C3),F3),A2)
         => ( ( C3 != zero_zero(A) )
           => sums(A,F3,divide_divide(A,A2,C3)) ) ) ) ).

% sums_mult_D
tff(fact_2639_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S2: A] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_dp(fun(nat,A),fun(nat,A),F3),S2)
           => sums(A,F3,S2) ) ) ) ).

% sums_Suc_imp
tff(fact_2640_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S2: A] :
          ( sums(A,aTP_Lamp_dp(fun(nat,A),fun(nat,A),F3),S2)
        <=> sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_2641_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_ef(fun(nat,A),fun(nat,A),F3),L)
         => sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_2642_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Nb: nat,F3: fun(nat,A),S2: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
             => ( aa(nat,A,F3,I2) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eg(nat,fun(fun(nat,A),fun(nat,A)),Nb),F3),S2)
          <=> sums(A,F3,S2) ) ) ) ).

% sums_zero_iff_shift
tff(fact_2643_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Mb: nat,Z: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_eh(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_2644_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : sums(A,aTP_Lamp_dr(fun(nat,A),fun(nat,A),A2),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_2645_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),Nb: nat,S2: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(nat,fun(nat,A)),F3),Nb),S2)
        <=> sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_iff_shift
tff(fact_2646_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),Nb: nat,S2: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(nat,fun(nat,A)),F3),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),S2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_ord_lessThan(nat,Nb))))
        <=> sums(A,F3,S2) ) ) ).

% sums_iff_shift'
tff(fact_2647_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S2: A,Nb: nat] :
          ( sums(A,F3,S2)
         => sums(A,aa(nat,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(nat,fun(nat,A)),F3),Nb),aa(A,A,aa(A,fun(A,A),minus_minus(A),S2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_split_initial_segment
tff(fact_2648_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G3: fun(nat,A),S3: A,A5: set(nat),S4: A,F3: fun(nat,A)] :
          ( sums(A,G3,S3)
         => ( aa(set(nat),$o,finite_finite(nat),A5)
           => ( ( S4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ei(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G3),F3)),A5)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ej(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G3),A5),F3),S4) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_2649_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_2650_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_2651_pi__half__neq__two,axiom,
    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_2652_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_2653_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( ! [X3: A,Xa2: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa2),R)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa2),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_2654_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_2655_pi__half__neq__zero,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_2656_pi__half__less__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(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_2657_pi__half__le__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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_2658_pi__half__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% pi_half_gt_zero
tff(fact_2659_pi__half__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% pi_half_ge_zero
tff(fact_2660_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_2661_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arctan,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% arctan_ubound
tff(fact_2662_arctan__one,axiom,
    aa(real,real,arctan,one_one(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) ).

% arctan_one
tff(fact_2663_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => sums(A,aa(A,fun(nat,A),power_power(A),C3),divide_divide(A,one_one(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C3))) ) ) ).

% geometric_sums
tff(fact_2664_power__half__series,axiom,
    sums(real,aTP_Lamp_ek(nat,real),one_one(real)) ).

% power_half_series
tff(fact_2665_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X4: A,Y3: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R2)
         => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),S2) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2) ) ).

% subrelI
tff(fact_2666_minus__pi__half__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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_2667_arctan__lbound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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_2668_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ).

% arctan_bounded
tff(fact_2669_sums__if_H,axiom,
    ! [G3: fun(nat,real),Xb: real] :
      ( sums(real,G3,Xb)
     => sums(real,aTP_Lamp_el(fun(nat,real),fun(nat,real),G3),Xb) ) ).

% sums_if'
tff(fact_2670_sums__if,axiom,
    ! [G3: fun(nat,real),Xb: real,F3: fun(nat,real),Y: real] :
      ( sums(real,G3,Xb)
     => ( sums(real,F3,Y)
       => sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_em(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G3),F3),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% sums_if
tff(fact_2671_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,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,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))))))))))) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) ).

% machin_Euler
tff(fact_2672_machin,axiom,
    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,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,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_2673_sin__cos__npi,axiom,
    ! [Nb: nat] : sin(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_2674_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C3: fun(nat,A),Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),C3),Xb))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_eo(fun(nat,A),fun(A,fun(nat,A)),C3),Xb),suminf(A,aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),C3),Xb))) ) ) ).

% diffs_equiv
tff(fact_2675_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : cos(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_2676_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M: nat,N: nat] :
              ( aa(nat,$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,X6,M)),aa(nat,A,X6,N)) )
         => topological_monoseq(A,X6) ) ) ).

% monoI1
tff(fact_2677_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M: nat,N: nat] :
              ( aa(nat,$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,X6,N)),aa(nat,A,X6,M)) )
         => topological_monoseq(A,X6) ) ) ).

% monoI2
tff(fact_2678_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [M5: nat,N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,M5)),aa(nat,A,X6,N4)) )
            | ! [M5: nat,N4: nat] :
                ( aa(nat,$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,X6,N4)),aa(nat,A,X6,M5)) ) ) ) ) ).

% monoseq_def
tff(fact_2679_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_2680_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_2681_cos__pi,axiom,
    cos(real,pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_2682_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_2683_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_2684_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_2685_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_2686_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_2687_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_2688_cos__pi__half,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_2689_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_2690_sin__pi__half,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_2691_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_2692_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_2693_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_2694_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_2695_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_2696_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_2697_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_2698_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_2699_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_2700_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_2701_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_2702_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(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_2703_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(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_2704_polar__Ex,axiom,
    ! [Xb: real,Y: real] :
    ? [R4: real,A4: real] :
      ( ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),R4),cos(real,A4)) )
      & ( Y = aa(real,real,aa(real,fun(real,real),times_times(real),R4),sin(real,A4)) ) ) ).

% polar_Ex
tff(fact_2705_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_2706_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_2707_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_2708_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_2709_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_2710_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_2711_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_2712_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_2713_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_2714_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_2715_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_2716_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_2717_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_2718_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_2719_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_2720_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_2721_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_2722_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)) = 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_2723_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)) = 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_2724_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)) = 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_2725_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,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,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_2726_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,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,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_2727_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,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,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_2728_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_2729_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_2730_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_2731_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C3: fun(nat,A),X: nat] : aa(nat,A,diffs(A,C3),X) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X))),aa(nat,A,C3,aa(nat,nat,suc,X))) ) ).

% diffs_def
tff(fact_2732_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_2733_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),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_2734_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),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),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_2735_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),Xb: A] :
          ( ! [X4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C3),X4))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C3),Xb)) ) ) ).

% termdiff_converges_all
tff(fact_2736_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_2737_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_2738_cos__is__zero,axiom,
    ? [X4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X4)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
      & ( cos(real,X4) = zero_zero(real) )
      & ! [Y4: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
            & ( cos(real,Y4) = zero_zero(real) ) )
         => ( Y4 = X4 ) ) ) ).

% cos_is_zero
tff(fact_2739_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_2740_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))
       => ? [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),pi)
            & ( cos(real,X4) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),pi)
                  & ( cos(real,Y4) = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% cos_total
tff(fact_2741_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),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_2742_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_2743_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_2744_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)) = 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_2745_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,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,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_2746_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),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_2747_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_2748_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_2749_sin__30,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_30
tff(fact_2750_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),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_2751_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),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),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),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),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_2752_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),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),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),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),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_2753_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),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),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_2754_cos__60,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_60
tff(fact_2755_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_2756_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_2757_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,K5: real,C3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xb)),K5)
         => ( ! [X4: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X4)),K5)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C3),X4)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_er(A,fun(fun(nat,A),fun(nat,A)),Xb),C3)) ) ) ) ).

% termdiff_converges
tff(fact_2758_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_2759_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(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_2760_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),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),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),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),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_2761_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),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),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_2762_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))
       => ? [X4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
            & ( sin(real,X4) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
                  & ( sin(real,Y4) = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% sin_total
tff(fact_2763_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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),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_2764_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(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),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_2765_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = one_one(real) )
    <=> ( ? [X3: 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),X3)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)
        | ? [X3: 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),X3)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_2766_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,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_2767_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] :
            ( 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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_2768_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ( ? [N4: nat] :
            ( 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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N4: nat] :
            ( 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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_2769_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] :
            ( ~ 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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_2770_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = zero_zero(real) )
    <=> ( ? [N4: nat] :
            ( ~ 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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N4: nat] :
            ( ~ 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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_2771_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),aa(nat,A,X6,aa(nat,nat,suc,N)))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI1
tff(fact_2772_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N))),aa(nat,A,X6,N))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI2
tff(fact_2773_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N4)),aa(nat,A,X6,aa(nat,nat,suc,N4)))
            | ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N4))),aa(nat,A,X6,N4)) ) ) ) ).

% monoseq_Suc
tff(fact_2774_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_es(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),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_2775_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_es(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),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_2776_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_es(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),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_2777_sin__paired,axiom,
    ! [Xb: real] : sums(real,aTP_Lamp_et(real,fun(nat,real),Xb),sin(real,Xb)) ).

% sin_paired
tff(fact_2778_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)) = 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_2779_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_2780_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_2781_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_2782_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_2783_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_2784_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_2785_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_2786_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_2787_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_2788_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_2789_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_2790_complex__diff,axiom,
    ! [A2: real,B2: real,C3: real,D3: real] : aa(complex,complex,aa(complex,fun(complex,complex),minus_minus(complex),complex2(A2,B2)),complex2(C3,D3)) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C3),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),D3)) ).

% complex_diff
tff(fact_2791_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_2792_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_2793_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_2794_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_2795_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_2796_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)
         => dvd_dvd(A,semiring_char_0_fact(A,Nb),semiring_char_0_fact(A,Mb)) ) ) ).

% fact_dvd
tff(fact_2797_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_2798_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_2799_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: nat,Nb: nat] : dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),Nb))) ) ).

% fact_fact_dvd_fact
tff(fact_2800_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_2801_complex__mult,axiom,
    ! [A2: real,B2: real,C3: real,D3: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),complex2(C3,D3)) = complex2(aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C3)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D3)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),D3)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),C3))) ).

% complex_mult
tff(fact_2802_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_2803_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_2804_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),X) = divide_divide(A,sin(A,X),cos(A,X)) ) ).

% tan_def
tff(fact_2805_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2))),semiring_char_0_fact(A,Nb)) ) ) ).

% choose_dvd
tff(fact_2806_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K2: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K2)),semiring_char_0_fact(A,pred_numeral(K2))) ) ).

% fact_numeral
tff(fact_2807_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_2808_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_2809_tan__45,axiom,
    aa(real,real,tan(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_2810_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_2811_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_2812_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),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_2813_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),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),X4)) ) ) ).

% lemma_tan_total
tff(fact_2814_tan__total,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X4)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( aa(real,real,tan(real),X4) = Y )
      & ! [Y4: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
            & ( aa(real,real,tan(real),Y4) = Y ) )
         => ( Y4 = X4 ) ) ) ).

% tan_total
tff(fact_2815_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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),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_2816_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),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),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),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),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_2817_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),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),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),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),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_2818_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X4)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( aa(real,real,tan(real),X4) = Y ) ) ).

% lemma_tan_total1
tff(fact_2819_tan__minus__45,axiom,
    aa(real,real,tan(real),aa(real,real,uminus_uminus(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_2820_tan__inverse,axiom,
    ! [Y: 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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y)) ).

% tan_inverse
tff(fact_2821_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)) = 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_2822_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),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_2823_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ? [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
          & ( aa(real,real,tan(real),X4) = Y ) ) ) ).

% tan_total_pos
tff(fact_2824_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(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_2825_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),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),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),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),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_2826_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),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),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_2827_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),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_2828_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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)),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_2829_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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),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_2830_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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),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_2831_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_eu(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xb),Diff)),set_ord_lessThan(nat,Nb)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_2832_Maclaurin__lemma,axiom,
    ! [H: real,F3: 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,F3,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_ev(real,fun(fun(nat,real),fun(nat,real)),H),J)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),B8),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_2833_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))) = 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_2834_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)) = 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_2835_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)) = 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_2836_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))
     => ? [Z3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))),Z3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))
          & ( aa(real,real,tan(real),Z3) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_2837_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tan(A),Xb) = 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_2838_cos__coeff__def,axiom,
    ! [X: nat] :
      cos_coeff(X) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,X,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),semiring_char_0_fact(real,X)),zero_zero(real)) ).

% cos_coeff_def
tff(fact_2839_cos__paired,axiom,
    ! [Xb: real] : sums(real,aTP_Lamp_ew(real,fun(nat,real),Xb),cos(real,Xb)) ).

% cos_paired
tff(fact_2840_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_ex(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),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_2841_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_ex(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),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_2842_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_ex(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),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_2843_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_ex(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),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_2844_sin__coeff__def,axiom,
    ! [X: nat] :
      sin_coeff(X) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X),zero_zero(real),divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),semiring_char_0_fact(real,X))) ).

% sin_coeff_def
tff(fact_2845_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))
            & ( exp(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ey(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(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_2846_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( exp(A,zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_2847_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( exp(real,Xb) = one_one(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_2848_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(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_2849_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(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_2850_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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_2851_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(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_2852_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_2853_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_2854_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [A5: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,A5)),A5) = aa(A,A,aa(A,fun(A,A),times_times(A),A5),exp(A,A5)) ) ).

% exp_times_arg_commute
tff(fact_2855_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_2856_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) )
         => ( 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),exp(A,Xb)),exp(A,Y)) ) ) ) ).

% exp_add_commuting
tff(fact_2857_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),exp(A,Xb)),exp(A,Y)) = exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ).

% mult_exp_exp
tff(fact_2858_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : exp(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),Y)) = divide_divide(A,exp(A,Xb),exp(A,Y)) ) ).

% exp_diff
tff(fact_2859_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_2860_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)),exp(real,Xb)) ) ).

% exp_gt_one
tff(fact_2861_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)),exp(real,Xb)) ).

% exp_ge_add_one_self
tff(fact_2862_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)
       => dvd_dvd(nat,Mb,semiring_char_0_fact(nat,Nb)) ) ) ).

% dvd_fact
tff(fact_2863_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),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_2864_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Nb: nat] : 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),exp(A,Xb)),Nb) ) ).

% exp_of_nat2_mult
tff(fact_2865_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xb: 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),exp(A,Xb)),Nb) ) ).

% exp_of_nat_mult
tff(fact_2866_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_2867_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)),exp(real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_2868_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y)
     => ? [X4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),one_one(real)))
          & ( exp(real,X4) = Y ) ) ) ).

% lemma_exp_total
tff(fact_2869_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(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),R2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_2870_ln__x__over__x__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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),divide_divide(real,aa(real,real,ln_ln(real),Y),Y)),divide_divide(real,aa(real,real,ln_ln(real),Xb),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_2871_exp__le,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% exp_le
tff(fact_2872_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),exp(A,divide_divide(A,Xb,aa(nat,A,semiring_1_of_nat(A),Nb)))),Nb) = exp(A,Xb) ) ) ) ).

% exp_divide_power_eq
tff(fact_2873_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tanh(A),Xb) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% tanh_altdef
tff(fact_2874_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(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_2875_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: 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),exp(A,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% exp_double
tff(fact_2876_sin__coeff__Suc,axiom,
    ! [Nb: nat] : sin_coeff(aa(nat,nat,suc,Nb)) = divide_divide(real,cos_coeff(Nb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb))) ).

% sin_coeff_Suc
tff(fact_2877_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% exp_bound_half
tff(fact_2878_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),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_2879_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),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),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_2880_cos__coeff__Suc,axiom,
    ! [Nb: nat] : cos_coeff(aa(nat,nat,suc,Nb)) = 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_2881_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)),divide_divide(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_2882_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)),divide_divide(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)))),Nb)),exp(real,aa(real,real,uminus_uminus(real),Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_2883_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,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_2884_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))
      & ( exp(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ey(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(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_2885_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)),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))))),exp(real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_2886_tanh__real__altdef,axiom,
    ! [Xb: real] : aa(real,real,tanh(real),Xb) = divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(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)),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_2887_sin__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => ( sin(real,Xb) = 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_2888_cos__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xb)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
     => ( cos(real,Xb) = 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_2889_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),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Nb)) ) ).

% pochhammer_double
tff(fact_2890_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_ez(A,A),Nb,zero_zero(A)) ) ).

% of_nat_code
tff(fact_2891_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fa(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Mb)) = aa(A,A,aa(A,fun(A,A),times_times(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,R2),aa(nat,nat,suc,Mb))) ) ).

% gchoose_row_sum_weighted
tff(fact_2892_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_2893_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_2894_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_2895_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,one_one(nat)) = A2 ) ).

% pochhammer_1
tff(fact_2896_gbinomial__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [K2: nat] : aa(nat,A,gbinomial(A,zero_zero(A)),aa(nat,nat,suc,K2)) = zero_zero(A) ) ).

% gbinomial_0(2)
tff(fact_2897_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_2898_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_2899_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_2900_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_2901_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_2902_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_2903_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_2904_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_2905_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_2906_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_2907_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_2908_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_2909_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_2910_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_2911_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_2912_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_2913_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_2914_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_2915_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_2916_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_2917_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_2918_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),K2) = 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))),K2)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A2),K2)),semiring_char_0_fact(A,K2)) ) ).

% gbinomial_pochhammer
tff(fact_2919_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),K2) = 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),K2))),one_one(A)),K2),semiring_char_0_fact(A,K2)) ) ).

% gbinomial_pochhammer'
tff(fact_2920_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_2921_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: 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,K2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A2),K2)),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2))) ) ).

% gbinomial_Suc_Suc
tff(fact_2922_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K2) = 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),K2)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_2923_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_2924_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Nb: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,Nb),I) = semiri8178284476397505188at_aux(A,Inc,Nb,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_2925_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I) = I ) ).

% of_nat_aux.simps(1)
tff(fact_2926_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_2927_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_2928_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2)) = 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,K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K2)) ) ).

% gbinomial_addition_formula
tff(fact_2929_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: 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),K2))),aa(nat,A,gbinomial(A,A2),K2)) = 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))),K2)) ) ).

% gbinomial_absorb_comp
tff(fact_2930_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K2: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K2)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),K2))),K2)),aa(nat,A,gbinomial(A,A2),K2)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_2931_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,A2),K2)) = 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),K2)),aa(nat,A,gbinomial(A,A2),K2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_mult_1
tff(fact_2932_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K2)),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),K2)),aa(nat,A,gbinomial(A,A2),K2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K2))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2)))) ) ).

% gbinomial_mult_1'
tff(fact_2933_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_2934_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_2935_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_2936_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_2937_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Nb: nat,K2: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K2) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2938_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K2) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2939_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K2) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2940_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_2941_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_2942_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_2943_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_2944_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: 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,K2))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K2))) = 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),K2)) ) ).

% Suc_times_gbinomial
tff(fact_2945_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: 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,K2))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2))) = 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))),K2)) ) ).

% gbinomial_absorption
tff(fact_2946_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Mb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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)),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K2)),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),K2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_2947_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_2948_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_2949_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_2950_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),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_2951_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_2952_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_2953_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_2954_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_2955_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C3: real,B2: real,D3: 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),C3)),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),D3)),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),C3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_2956_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_2957_cos__45,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = 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_2958_sin__45,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = 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_2959_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: 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,K2)) = aa(A,A,aa(A,fun(A,A),times_times(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,K2)))),aa(nat,A,gbinomial(A,A2),K2)) ) ).

% gbinomial_factors
tff(fact_2960_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: 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,K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K2)),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,K2)))) ) ).

% gbinomial_rec
tff(fact_2961_tan__60,axiom,
    aa(real,real,tan(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_2962_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: 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))),K2)),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))),K2)) = 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),K2))),one_one(A))),Nb)) ) ).

% gbinomial_index_swap
tff(fact_2963_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),K2) = 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))),K2)),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),K2)),A2)),one_one(A))),K2)) ) ).

% gbinomial_negated_upper
tff(fact_2964_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(nat,A,semiring_1_of_nat(A),K2))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K2)) ) ).

% pochhammer_absorb_comp
tff(fact_2965_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_2966_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_2967_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_2968_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( 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))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_2969_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_2970_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_2971_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_2972_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)) = 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_2973_cos__30,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = 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_2974_sin__60,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = 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_2975_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A2)),K2) = 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))),K2)),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),K2))),one_one(A))),K2)) ) ).

% gbinomial_minus
tff(fact_2976_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_2977_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
         => ( aa(nat,A,gbinomial(A,A2),K2) = 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),K2),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),K2)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_2978_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K2: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K2) = 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))),K2)),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),K2))),one_one(A)),K2)) ) ).

% pochhammer_minus
tff(fact_2979_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K2: 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),K2))),one_one(A)),K2) = 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))),K2)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K2)) ) ).

% pochhammer_minus'
tff(fact_2980_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_2981_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_2982_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( 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),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_2983_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))),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_2984_tan__30,axiom,
    aa(real,real,tan(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = 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_2985_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fb(nat,fun(nat,A),K2)),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),K2),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_2986_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),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_2987_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)),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)),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_2988_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_2989_cos__arctan,axiom,
    ! [Xb: real] : cos(real,aa(real,real,arctan,Xb)) = 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_2990_sin__arctan,axiom,
    ! [Xb: real] : sin(real,aa(real,real,arctan,Xb)) = 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_2991_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
         => ( aa(nat,A,gbinomial(A,A2),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),K2))),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),K2),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_2992_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),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),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_2993_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_2994_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,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_2995_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,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_2996_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] :
          aa(nat,A,gbinomial(A,A2),K2) = $ite(K2 = zero_zero(nat),one_one(A),divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_fc(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),one_one(nat)),one_one(A)),semiring_char_0_fact(A,K2))) ) ).

% gbinomial_code
tff(fact_2997_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),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_fd(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_2998_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_fe(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_2999_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_3000_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_fa(A,fun(nat,A),A2)),set_ord_atMost(nat,Mb)) = aa(A,A,aa(A,fun(A,A),times_times(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_3001_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K2: A] :
          ( member(A,I,set_ord_atMost(A,K2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),K2) ) ) ).

% atMost_iff
tff(fact_3002_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ff(B,A)),A5) = one_one(A) ) ).

% prod.neutral_const
tff(fact_3003_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_3004_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_3005_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,Xb)),set_ord_atMost(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% atMost_subset_iff
tff(fact_3006_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( 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_fg(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_3007_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( 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_fh(A,fun(fun(A,B),fun(A,B)),A2),B2)),S3) = $ite(member(A,A2,S3),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_3008_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),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atMost(A,H2))
        <=> ( ~ aa(A,$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_3009_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),set_ord_atMost(nat,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) ) ).

% sum.atMost_Suc
tff(fact_3010_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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),G3),set_ord_lessThan(nat,Nb))),aa(nat,A,G3,Nb)) ) ).

% prod.lessThan_Suc
tff(fact_3011_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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),G3),set_ord_atMost(nat,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) ) ).

% prod.atMost_Suc
tff(fact_3012_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb)))) ) ).

% prod.cl_ivl_Suc
tff(fact_3013_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),A5: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) != one_one(A) )
         => ~ ! [A4: B] :
                ( member(B,A4,A5)
               => ( aa(B,A,G3,A4) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_3014_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A5)
             => ( aa(A,B,G3,X4) = one_one(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_3015_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),H: fun(B,A),A5: 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_fi(fun(B,A),fun(fun(B,A),fun(B,A)),G3),H)),A5) = 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),G3),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),A5)) ) ).

% prod.distrib
tff(fact_3016_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F3: fun(B,A),G3: fun(B,A),A5: 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_fj(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),A5) = divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5)) ) ).

% prod_dividef
tff(fact_3017_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A5: 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),F3),A5)),Nb) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(nat,fun(B,A),aTP_Lamp_fk(fun(B,A),fun(nat,fun(B,A)),F3),Nb)),A5) ) ).

% prod_power_distrib
tff(fact_3018_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F3: fun(B,A),A2: A,A5: 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_bf(fun(B,A),fun(A,fun(B,A)),F3),A2)),A5),A2) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5),A2) ) ).

% mod_prod_eq
tff(fact_3019_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G3)),set_ord_atMost(nat,Nb))) ) ).

% prod.atMost_Suc_shift
tff(fact_3020_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_fm(fun(nat,fun(nat,A)),fun(nat,A),A2)),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_fo(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_ord_lessThan(nat,Nb)) ) ).

% prod.nested_swap'
tff(fact_3021_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fp(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_3022_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),aa(A,B,G3,I2)) ) )
         => aa(B,$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),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)) ) ) ).

% prod_mono
tff(fact_3023_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X4)) )
         => aa(B,$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),F3),A5)) ) ) ).

% prod_nonneg
tff(fact_3024_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,X4)) )
         => aa(B,$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),F3),A5)) ) ) ).

% prod_pos
tff(fact_3025_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F3,X4)) )
         => aa(B,$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),F3),A5)) ) ) ).

% prod_ge_1
tff(fact_3026_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F3: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F3),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fq(fun(nat,A),fun(nat,fun(A,A)),F3),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_3027_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Nb))) ) ).

% prod.atMost_shift
tff(fact_3028_lessThan__Suc__atMost,axiom,
    ! [K2: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K2)) = set_ord_atMost(nat,K2) ).

% lessThan_Suc_atMost
tff(fact_3029_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_fr(set(A),fun(fun(A,$o),fun(A,$o)),A5),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_fs(fun(A,B),fun(fun(A,$o),fun(A,B)),G3),P)),A5) ) ) ) ).

% prod.inter_filter
tff(fact_3030_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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_fl(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_3031_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C3: A,F3: fun(B,nat),A5: set(B)] : aa(nat,A,aa(A,fun(nat,A),power_power(A),C3),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F3),A5)) = 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_ft(A,fun(fun(B,nat),fun(B,A)),C3),F3)),A5) ) ).

% power_sum
tff(fact_3032_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fu(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_3033_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,A5)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X4))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),one_one(B)) ) )
         => aa(B,$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),F3),A5)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_3034_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G3: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X15: A,Y15: A,X22: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X22)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,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)) )
           => ( aa(set(B),$o,finite_finite(B),S3)
             => ( ! [X4: B] :
                    ( member(B,X4,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X4)),aa(B,A,G3,X4)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),S3)) ) ) ) ) ) ).

% prod.related
tff(fact_3035_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T3: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T2: set(B),G3: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S4)
         => ( aa(set(B),$o,finite_finite(B),T3)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4))
                   => 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)) )
               => ( ! [B4: B] :
                      ( member(B,B4,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,I,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
                       => member(A,aa(B,A,I,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S4)
                         => ( aa(A,C,G3,A4) = one_one(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T3)
                           => ( aa(B,C,H,B4) = one_one(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S3)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G3,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G3),S3) = 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_3036_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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_fv(fun(nat,A),fun(nat,A),G3)),set_ord_atMost(nat,Nb)) ) ).

% prod.in_pairs_0
tff(fact_3037_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_fw(fun(A,B),fun(A,$o),G3)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3038_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: 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_fx(fun(nat,A),fun(nat,fun(nat,A)),G3),Nb)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_lessThan(nat,Nb)) ) ).

% prod.nat_diff_reindex
tff(fact_3039_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,A2)),set_ord_lessThan(A,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3040_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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_fy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or1337092689740270186AtMost(nat,Nb,Mb)) ) ).

% prod.atLeastAtMost_rev
tff(fact_3041_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K2: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$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),K2),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_fz(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),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_ga(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),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_3042_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),I: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I6)
         => ( member(A,I,I6)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F3,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I6)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F3,I2)) )
               => 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),F3),I6)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3043_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I6: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I6)
         => ( ( I6 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I6)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F3,I2)) )
             => 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),F3),I6)) ) ) ) ) ).

% less_1_prod
tff(fact_3044_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(A),A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( aa(set(A),$o,finite_finite(A),A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3045_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S3: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,G3,X4) = one_one(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S3)
                   => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3046_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S3: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,H,X4) = one_one(B) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,S3)
                   => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S3) = 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_3047_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S3: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,G3,X4) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),T2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S3) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3048_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S3: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,G3,X4) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),T2) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3049_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A5))
                   => ( aa(A,B,G3,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B5))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),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),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B5) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3050_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),A5))
                   => ( aa(A,B,G3,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C5),B5))
                     => ( aa(A,B,H,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B5) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),C5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3051_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_3052_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$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),G3),set_or1337092689740270186AtMost(nat,Mb,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,aa(nat,nat,suc,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3053_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$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),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3054_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Nb))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3055_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$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),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(nat,A,G3,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3056_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),G3)),set_ord_atMost(nat,Nb))) ) ).

% sum.atMost_Suc_shift
tff(fact_3057_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),I: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),F3)),set_ord_atMost(nat,I)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_3058_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Nb: nat,D3: fun(nat,A)] :
          ( ! [X3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),C3),X3)),set_ord_atMost(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),D3),X3)),set_ord_atMost(nat,Nb))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
             => ( aa(nat,A,C3,I3) = aa(nat,A,D3,I3) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_3059_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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_fl(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Nb)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3060_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B5: A] :
          ( ! [N: nat] : aa(A,$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),set_ord_atMost(nat,N))),B5)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_3061_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_ca(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb))) ) ).

% fact_prod
tff(fact_3062_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_gc(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_ord_atMost(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ge(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_ord_lessThan(nat,Nb)) ) ).

% sum.nested_swap'
tff(fact_3063_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ! [I2: A] :
                ( member(A,I2,A5)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,I2)),aa(A,B,G3,I2)) ) )
           => ( ( A5 != 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),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)) ) ) ) ) ).

% prod_mono_strict
tff(fact_3064_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( 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),F3),A5))
          <=> ? [X3: A] :
                ( member(A,X3,A5)
                & dvd_dvd(B,aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)),aa(A,B,F3,X3)) ) ) ) ) ).

% even_prod_iff
tff(fact_3065_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A),P2: nat] :
          ( aa(nat,$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),G3),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),G3),set_or1337092689740270186AtMost(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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_3066_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_3067_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,F3,A2,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A2),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_3068_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I6: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,I6)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z,I2))),one_one(real)) )
         => ( ! [I2: A] :
                ( member(A,I2,I6)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W,I2))),one_one(real)) )
           => aa(real,$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),I6)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),W),I6)))),aa(set(A),real,groups7311177749621191930dd_sum(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_gf(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I6)) ) ) ) ).

% norm_prod_diff
tff(fact_3069_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Nb: nat] :
          ( ! [X3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),C3),X3)),set_ord_atMost(nat,Nb)) = zero_zero(A)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
             => ( aa(nat,A,C3,I3) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_3070_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C3: fun(nat,A),Nb: nat,K2: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(fun(nat,A),fun(A,fun(nat,A)),C3),W2)),set_ord_atMost(nat,Nb)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => ( aa(nat,A,C3,K2) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_3071_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Nb))) ) ).

% sum.atMost_shift
tff(fact_3072_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),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,F3),set_ord_atMost(nat,Mb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),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_3073_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gh(A,fun(nat,A),A2)),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_3074_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$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,$o),aTP_Lamp_gi(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gk(fun(nat,fun(nat,A)),fun(nat,A),G3)),set_ord_atMost(nat,Nb)) ) ).

% sum.triangle_reindex_eq
tff(fact_3075_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_ca(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_3076_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F3,B4)) )
             => ( ! [A4: A] :
                    ( member(A,A4,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,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),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B5)) ) ) ) ) ) ).

% prod_mono2
tff(fact_3077_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)),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_3078_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Nb: nat] :
          ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_gl(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb)))
        <=> ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
              & ( aa(nat,A,C3,I3) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_3079_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K2: nat,Nb: nat] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_gl(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb))) ) ) ) ).

% polyfun_roots_finite
tff(fact_3080_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_gm(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod
tff(fact_3081_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: fun(nat,A),A2: A,Nb: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),C3),A2)),set_ord_atMost(nat,Nb)) = zero_zero(A) )
         => ~ ! [B4: fun(nat,A)] :
                ~ ! [Z4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),C3),Z4)),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),Z4),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),set_ord_lessThan(nat,Nb))) ) ) ).

% polyfun_linear_factor_root
tff(fact_3082_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: fun(nat,A),Nb: nat,A2: A] :
        ? [B4: fun(nat,A)] :
        ! [Z4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),C3),Z4)),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),Z4),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),set_ord_lessThan(nat,Nb)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),C3),A2)),set_ord_atMost(nat,Nb))) ) ).

% polyfun_linear_factor
tff(fact_3083_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_go(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)) ) ).

% pochhammer_prod_rev
tff(fact_3084_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)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)))) ) ) ) ).

% sum_power_shift
tff(fact_3085_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$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,$o),aTP_Lamp_gp(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_gk(fun(nat,fun(nat,A)),fun(nat,A),G3)),set_ord_lessThan(nat,Nb)) ) ).

% sum.triangle_reindex
tff(fact_3086_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( 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_ca(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_3087_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_gq(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_gq(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_3088_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_gq(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_gq(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_gs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_3089_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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_fv(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.in_pairs
tff(fact_3090_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_gt(fun(nat,A),fun(nat,fun(A,A)),F3),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3091_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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_bz(fun(nat,A),fun(nat,A),G3)),set_ord_atMost(nat,Nb)) ) ).

% sum.in_pairs_0
tff(fact_3092_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Mb: nat,A2: fun(nat,A),Nb: nat,B2: fun(nat,A),Xb: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),I2)
             => ( aa(nat,A,A2,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$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_gn(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),set_ord_atMost(nat,Mb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),B2),Xb)),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_gv(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ) ) ) ).

% polynomial_product
tff(fact_3093_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_go(A,fun(nat,fun(nat,A)),A2),Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3094_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Nb: nat,K2: A] :
          ( ! [X3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),C3),X3)),set_ord_atMost(nat,Nb)) = K2
        <=> ( ( aa(nat,A,C3,zero_zero(nat)) = K2 )
            & ! [X3: nat] :
                ( member(nat,X3,set_or1337092689740270186AtMost(nat,one_one(nat),Nb))
               => ( aa(nat,A,C3,X3) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_3095_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_gw(A,fun(nat,A),A2)),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_3096_polynomial__product__nat,axiom,
    ! [Mb: nat,A2: fun(nat,nat),Nb: nat,B2: fun(nat,nat),Xb: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),I2)
         => ( aa(nat,nat,A2,I2) = 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_gx(fun(nat,nat),fun(nat,fun(nat,nat)),A2),Xb)),set_ord_atMost(nat,Mb))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gx(fun(nat,nat),fun(nat,fun(nat,nat)),B2),Xb)),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_gz(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_3097_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_gq(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_gq(fun(nat,A),fun(nat,real),B2))
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gs(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_3098_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K2)) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ha(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K2)),semiring_char_0_fact(A,aa(nat,nat,suc,K2))) ) ).

% gbinomial_Suc
tff(fact_3099_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K2: nat,G3: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$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),K2),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_hb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),set_ord_atMost(nat,P2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K2),G3),H)),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_3100_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_hd(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),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_he(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),set_ord_atMost(nat,Mb)) ) ).

% gbinomial_partial_sum_poly
tff(fact_3101_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_hf(nat,fun(A,fun(A,fun(nat,A))),Nb),Z),A2)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_3102_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)),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))),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_3103_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hg(nat,fun(nat,A),Mb)),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_3104_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_hd(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),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_hh(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Mb),A2),Xb),Y)),set_ord_atMost(nat,Mb)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_3105_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_gn(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),set_ord_atMost(nat,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),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_hj(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff_alt
tff(fact_3106_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E3: real,C3: fun(nat,A),Nb: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => ? [M7: real] :
            ! [Z4: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M7),real_V7770717601297561774m_norm(A,Z4))
             => aa(real,$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_ct(fun(nat,A),fun(A,fun(nat,A)),C3),Z4)),set_ord_atMost(nat,Nb)))),aa(real,real,aa(real,fun(real,real),times_times(real),E3),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z4)),aa(nat,nat,suc,Nb)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_3107_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_gn(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),set_ord_atMost(nat,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),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_hl(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff
tff(fact_3108_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_3109_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)))),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_3110_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_hm(nat,fun(nat,A),Nb)),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_3111_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_hn(nat,fun(nat,A),Nb)),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_3112_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_3113_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_3114_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_3115_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_3116_binomial__n__n,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),Nb) = one_one(nat) ).

% binomial_n_n
tff(fact_3117_binomial__0__Suc,axiom,
    ! [K2: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K2)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_3118_binomial__1,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(nat,nat,suc,zero_zero(nat))) = Nb ).

% binomial_1
tff(fact_3119_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( ( aa(nat,nat,binomial(Nb),K2) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2) ) ).

% binomial_eq_0_iff
tff(fact_3120_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K2))) ).

% binomial_Suc_Suc
tff(fact_3121_binomial__n__0,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_3122_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3123_prod__eq__1__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A5) = one_one(nat) )
      <=> ! [X3: A] :
            ( member(A,X3,A5)
           => ( aa(A,nat,F3,X3) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_3124_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb) ) ).

% zero_less_binomial_iff
tff(fact_3125_arccos__minus__1,axiom,
    aa(real,real,arccos,aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_3126_prod__pos__nat__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( 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),F3),A5))
      <=> ! [X3: A] :
            ( member(A,X3,A5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X3)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3127_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_3128_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_3129_arccos__0,axiom,
    aa(real,real,arccos,zero_zero(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% arccos_0
tff(fact_3130_arcsin__1,axiom,
    aa(real,real,arcsin,one_one(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% arcsin_1
tff(fact_3131_arcsin__minus__1,axiom,
    aa(real,real,arcsin,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% arcsin_minus_1
tff(fact_3132_choose__one,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),one_one(nat)) = Nb ).

% choose_one
tff(fact_3133_binomial__eq__0,axiom,
    ! [Nb: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2)
     => ( aa(nat,nat,binomial(Nb),K2) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_3134_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K2))),aa(nat,nat,suc,K2)) ).

% Suc_times_binomial_eq
tff(fact_3135_Suc__times__binomial,axiom,
    ! [K2: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K2)) ).

% Suc_times_binomial
tff(fact_3136_binomial__symmetric,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
     => ( aa(nat,nat,binomial(Nb),K2) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)) ) ) ).

% binomial_symmetric
tff(fact_3137_choose__mult__lemma,axiom,
    ! [Mb: nat,R2: nat,K2: 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),R2)),K2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2)),K2)) = 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),R2)),K2)),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_3138_binomial__le__pow,axiom,
    ! [R2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),R2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R2)) ) ).

% binomial_le_pow
tff(fact_3139_zero__less__binomial,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(Nb),K2)) ) ).

% zero_less_binomial
tff(fact_3140_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_3141_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K2)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,suc,K2)) ).

% binomial_Suc_Suc_eq_times
tff(fact_3142_choose__mult,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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),K2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),K2))) ) ) ) ).

% choose_mult
tff(fact_3143_binomial__absorb__comp,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)),aa(nat,nat,binomial(Nb),K2)) = 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))),K2)) ).

% binomial_absorb_comp
tff(fact_3144_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ho(nat,fun(nat,nat),Mb)),set_ord_atMost(nat,Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,Mb)) ).

% sum_choose_upper
tff(fact_3145_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_3146_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_3147_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_3148_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_3149_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_3150_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_3151_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_3152_binomial__absorption,axiom,
    ! [K2: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K2)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K2))) = 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))),K2)) ).

% binomial_absorption
tff(fact_3153_binomial__fact__lemma,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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,K2)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)))),aa(nat,nat,binomial(Nb),K2)) = semiring_char_0_fact(nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_3154_prod__int__plus__eq,axiom,
    ! [I: 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,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_ax(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).

% prod_int_plus_eq
tff(fact_3155_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hp(nat,fun(nat,nat),R2)),set_ord_atMost(nat,Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R2),Nb))),Nb) ).

% sum_choose_lower
tff(fact_3156_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hq(nat,fun(nat,nat),Nb)),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_3157_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hq(nat,fun(nat,nat),Nb)),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_3158_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Nb),aa(nat,A,semiring_1_of_nat(A),K2))),K2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K2))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_3159_binomial__mono,axiom,
    ! [K2: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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),K2)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_mono
tff(fact_3160_binomial__maximum_H,axiom,
    ! [Nb: nat,K2: 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)),K2)),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_3161_binomial__maximum,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(Nb),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% binomial_maximum
tff(fact_3162_binomial__antimono,axiom,
    ! [K2: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),K2)
       => ( 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),K2)) ) ) ) ).

% binomial_antimono
tff(fact_3163_binomial__le__pow2,axiom,
    ! [Nb: nat,K2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K2)),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_3164_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_3165_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_3166_choose__reduce__nat,axiom,
    ! [Nb: nat,K2: 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)),K2)
       => ( aa(nat,nat,binomial(Nb),K2) = 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),K2),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K2)) ) ) ) ).

% choose_reduce_nat
tff(fact_3167_times__binomial__minus1__eq,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(nat,nat,binomial(Nb),K2)) = 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),K2),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_3168_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_3169_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_3170_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_3171_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_3172_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_3173_binomial__altdef__nat,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
     => ( aa(nat,nat,binomial(Nb),K2) = 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,K2)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)))) ) ) ).

% binomial_altdef_nat
tff(fact_3174_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_hr(nat,fun(nat,fun(nat,nat)),Mb),Nb)),set_ord_atMost(nat,Mb)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_3175_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hs(nat,fun(nat,fun(nat,fun(nat,nat))),Mb),Nb),R2)),set_ord_atMost(nat,R2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),R2) ).

% vandermonde
tff(fact_3176_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),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),K2)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K2))) ) ).

% binomial_less_binomial_Suc
tff(fact_3177_binomial__strict__mono,axiom,
    ! [K2: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),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),K2)),aa(nat,nat,binomial(Nb),K7)) ) ) ).

% binomial_strict_mono
tff(fact_3178_binomial__strict__antimono,axiom,
    ! [K2: nat,K7: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),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))),K2))
       => ( 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),K2)) ) ) ) ).

% binomial_strict_antimono
tff(fact_3179_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
     => ( aa(nat,nat,binomial(Nb),aa(nat,nat,suc,divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = aa(nat,nat,binomial(Nb),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% central_binomial_odd
tff(fact_3180_binomial__addition__formula,axiom,
    ! [Nb: nat,K2: 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,K2)) = 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,K2))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),K2)) ) ) ).

% binomial_addition_formula
tff(fact_3181_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K2))) = 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),K2))) ) ) ) ).

% fact_binomial
tff(fact_3182_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K2)) = 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,K2)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)))) ) ) ) ).

% binomial_fact
tff(fact_3183_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_3184_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_3185_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_3186_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_3187_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_3188_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$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,$o),aTP_Lamp_gi(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hu(fun(nat,fun(nat,A)),fun(nat,A),G3)),set_ord_atMost(nat,Nb)) ) ).

% prod.triangle_reindex_eq
tff(fact_3189_choose__row__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(Nb)),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_3190_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_hv(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ).

% binomial
tff(fact_3191_choose__two,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = 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_3192_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_hw(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ) ).

% binomial_ring
tff(fact_3193_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_hx(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ) ).

% pochhammer_binomial_sum
tff(fact_3194_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_3195_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$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,$o),aTP_Lamp_gp(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hu(fun(nat,fun(nat,A)),fun(nat,A),G3)),set_ord_lessThan(nat,Nb)) ) ).

% prod.triangle_reindex
tff(fact_3196_choose__square__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_hy(nat,fun(nat,nat),Nb)),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_3197_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_3198_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_hz(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3199_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)))),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_3200_choose__linear__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ia(nat,fun(nat,nat),Nb)),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_3201_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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ).

% arccos_le_pi2
tff(fact_3202_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),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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3203_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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_3204_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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ).

% arcsin_ubound
tff(fact_3205_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),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)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_3206_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(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),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_3207_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_ib(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3208_binomial__code,axiom,
    ! [Nb: nat,K2: nat] :
      aa(nat,nat,binomial(Nb),K2) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K2),
        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))),K2)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2)),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),K2)),one_one(nat)),Nb,one_one(nat)),semiring_char_0_fact(nat,K2))) ) ).

% binomial_code
tff(fact_3209_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),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)),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_3210_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),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_3211_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),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),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_3212_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),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),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_3213_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),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_3214_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_id(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_3215_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_ex(real,fun(nat,real),Xb)),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_3216_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_if(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_3217_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_3218_cos__npi__int,axiom,
    ! [Nb: int] :
      cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Nb))) = $ite(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_3219_of__nat__id,axiom,
    ! [Nb: nat] : aa(nat,nat,semiring_1_of_nat(nat),Nb) = Nb ).

% of_nat_id
tff(fact_3220_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),real_V8093663219630862766scaleR(A,A2,Y)) = real_V8093663219630862766scaleR(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) ) ).

% mult_scaleR_right
tff(fact_3221_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),real_V8093663219630862766scaleR(A,A2,Xb)),Y) = real_V8093663219630862766scaleR(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) ) ).

% mult_scaleR_left
tff(fact_3222_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_3223_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_3224_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_3225_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),divide_divide(A,A2,B2)) = divide_divide(A,B2,A2) ) ).

% inverse_divide
tff(fact_3226_scaleR__one,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : real_V8093663219630862766scaleR(A,one_one(real),Xb) = Xb ) ).

% scaleR_one
tff(fact_3227_scaleR__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : real_V8093663219630862766scaleR(A,A2,real_V8093663219630862766scaleR(A,B2,Xb)) = real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2),Xb) ) ).

% scaleR_scaleR
tff(fact_3228_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_3229_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_3230_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_3231_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_3232_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_3233_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_3234_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% of_int_le_iff
tff(fact_3235_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,Nb: num] :
          ( ( aa(int,A,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_3236_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: num] : aa(int,A,ring_1_of_int(A),aa(num,int,numeral_numeral(int),K2)) = aa(num,A,numeral_numeral(A),K2) ) ).

% of_int_numeral
tff(fact_3237_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),real_V8093663219630862766scaleR(A,U,A2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),real_V8093663219630862766scaleR(A,U,B2)) )
        <=> ( ( A2 = B2 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_3238_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W),Z) ) ) ).

% of_int_less_iff
tff(fact_3239_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = one_one(A) )
        <=> ( Z = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_3240_of__int__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),one_one(int)) = one_one(A) ) ) ).

% of_int_1
tff(fact_3241_of__int__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_mult
tff(fact_3242_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_add
tff(fact_3243_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),real_V8093663219630862766scaleR(A,Xb,Y)),Nb) = 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_3244_of__int__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),minus_minus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_diff
tff(fact_3245_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,Nb: nat] : aa(int,A,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),aa(int,A,ring_1_of_int(A),Z)),Nb) ) ).

% of_int_power
tff(fact_3246_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),aa(int,A,ring_1_of_int(A),B2)),W) = aa(int,A,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_3247_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: int,B2: int,W: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Xb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,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_3248_dbl__dec__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K2)) = aa(num,A,numeral_numeral(A),bitM(K2)) ) ).

% dbl_dec_simps(5)
tff(fact_3249_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_3250_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_3251_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_3252_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_3253_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)) = divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),W)) ) ).

% inverse_eq_divide_numeral
tff(fact_3254_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: 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_3255_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),U),A2)),real_V8093663219630862766scaleR(A,U,A2)) = A2 ) ).

% scaleR_collapse
tff(fact_3256_norm__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: real,Xb: A] : real_V7770717601297561774m_norm(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_3257_pred__numeral__simps_I2_J,axiom,
    ! [K2: num] : pred_numeral(aa(num,num,bit0,K2)) = aa(num,nat,numeral_numeral(nat),bitM(K2)) ).

% pred_numeral_simps(2)
tff(fact_3258_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))) = 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_3259_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_3260_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_3261_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_3262_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_3263_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$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_3264_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$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_3265_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_less_numeral_iff
tff(fact_3266_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_3267_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_3268_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_3269_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,A2: 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)) = 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_3270_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_3271_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_3272_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,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$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_3273_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,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W))
        <=> aa(int,$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_3274_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),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_3275_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) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_3276_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,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$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_3277_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,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W))
        <=> aa(int,$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_3278_sin__npi__int,axiom,
    ! [Nb: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Nb))) = zero_zero(real) ).

% sin_npi_int
tff(fact_3279_tan__periodic__int,axiom,
    ! [Xb: real,I: 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),aa(int,real,ring_1_of_int(real),I)),pi))) = aa(real,real,tan(real),Xb) ).

% tan_periodic_int
tff(fact_3280_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,A2: A] : real_V8093663219630862766scaleR(A,divide_divide(real,one_one(real),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)) = real_V8093663219630862766scaleR(A,divide_divide(real,aa(num,real,numeral_numeral(real),W),aa(num,real,numeral_numeral(real),V)),A2) ) ).

% inverse_scaleR_times
tff(fact_3281_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,A2: A] : real_V8093663219630862766scaleR(A,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)) = real_V8093663219630862766scaleR(A,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_3282_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A] : real_V8093663219630862766scaleR(A,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_3283_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,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$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_3284_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,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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_3285_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,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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_3286_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,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$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_3287_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),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_3288_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) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_3289_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)),aa(int,real,ring_1_of_int(real),Nb))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_3290_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)),aa(int,real,ring_1_of_int(real),Nb))) = one_one(real) ).

% cos_int_2pin
tff(fact_3291_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,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$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_3292_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,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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_3293_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,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$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_3294_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,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$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_3295_real__scaleR__def,axiom,
    ! [A2: real,Xb: real] : real_V8093663219630862766scaleR(real,A2,Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),Xb) ).

% real_scaleR_def
tff(fact_3296_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),aa(int,A,ring_1_of_int(A),Xb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),Xb)) ) ).

% mult_of_int_commute
tff(fact_3297_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_3298_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_less_of_int
tff(fact_3299_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),Xb) ) ).

% ex_of_int_less
tff(fact_3300_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_le_of_int
tff(fact_3301_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),aa(int,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),aa(int,A,ring_1_of_int(A),Xaa))) ) ).

% mult_inverse_of_int_commute
tff(fact_3302_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_3303_scaleR__right__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Y: 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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,A2,Y)) ) ).

% scaleR_right_diff_distrib
tff(fact_3304_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Y: 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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,A2,Y)) ) ).

% scaleR_right_distrib
tff(fact_3305_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_3306_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Xb: A,C3: A,Y: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,Mb,Xb)),C3) = Y )
          <=> ( Xb = aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb),Y)),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb),C3)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_3307_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Mb: real,Y: A,Xb: A,C3: A] :
          ( ( Mb != zero_zero(real) )
         => ( ( Y = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,Mb,Xb)),C3) )
          <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb),Y)),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),Mb),C3)) = Xb ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_3308_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),real_V8093663219630862766scaleR(A,C3,A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_3309_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C3,A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_3310_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C3,A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_3311_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),real_V8093663219630862766scaleR(A,C3,A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_3312_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),real_V8093663219630862766scaleR(A,C3,A2)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_3313_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,C3,A2)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_3314_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,C3,A2)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_3315_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),real_V8093663219630862766scaleR(A,C3,A2)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_3316_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_ig(A,fun(nat,A),Xb)) ) ).

% summable_exp_generic
tff(fact_3317_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C3,A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3318_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C3,A2)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3319_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C3,A2)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3320_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C3,A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3321_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,C3,A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_3322_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C3,A2)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_3323_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),real_V8093663219630862766scaleR(A,C3,A2)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_3324_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),real_V8093663219630862766scaleR(A,C3,A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_3325_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_3326_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_3327_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_3328_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_3329_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_3330_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_3331_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_3332_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_3333_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_3334_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_3335_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_3336_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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_3337_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: 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_3338_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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_3339_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),A2) = divide_divide(A,one_one(A),A2) ) ).

% inverse_eq_divide
tff(fact_3340_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_3341_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_3342_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_3343_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: 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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ).

% scaleR_left_distrib
tff(fact_3344_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: 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),real_V8093663219630862766scaleR(A,Xb,Xaa)),real_V8093663219630862766scaleR(A,Y,Xaa)) ) ).

% scaleR_left.add
tff(fact_3345_divide__real__def,axiom,
    ! [Xb: real,Y: 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_3346_scaleR__left__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: 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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ).

% scaleR_left_diff_distrib
tff(fact_3347_scaleR__left_Odiff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: 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),real_V8093663219630862766scaleR(A,Xb,Xaa)),real_V8093663219630862766scaleR(A,Y,Xaa)) ) ).

% scaleR_left.diff
tff(fact_3348_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ig(A,fun(nat,A),Xb),exp(A,Xb)) ) ).

% exp_converges
tff(fact_3349_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : exp(A,X) = suminf(A,aTP_Lamp_ig(A,fun(nat,A),X)) ) ).

% exp_def
tff(fact_3350_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_ih(A,fun(nat,real),Xb)) ) ).

% summable_norm_exp
tff(fact_3351_complex__scaleR,axiom,
    ! [R2: real,A2: real,B2: real] : real_V8093663219630862766scaleR(complex,R2,complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),A2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),B2)) ).

% complex_scaleR
tff(fact_3352_semiring__norm_I27_J,axiom,
    ! [Nb: num] : bitM(aa(num,num,bit0,Nb)) = aa(num,num,bit1,bitM(Nb)) ).

% semiring_norm(27)
tff(fact_3353_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_3354_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_3355_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_3356_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_3357_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_3358_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_3359_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_3360_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_3361_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_3362_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_3363_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_3364_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_3365_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = divide_divide(A,one_one(A),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3366_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,C3)),real_V8093663219630862766scaleR(A,B2,C3)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_3367_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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ) ) ).

% scaleR_right_mono
tff(fact_3368_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C3,A2)),real_V8093663219630862766scaleR(A,C3,B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_3369_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C3,A2)),real_V8093663219630862766scaleR(A,C3,B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_3370_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C3,A2)),real_V8093663219630862766scaleR(A,C3,B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_3371_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C3: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,C3,A2)),real_V8093663219630862766scaleR(A,C3,B2)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_3372_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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,A2,Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_3373_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C3: A,B2: real,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A2,E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,B2,E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2),E3)),D3)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3374_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C3: A,B2: real,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,A2,E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,B2,E3)),D3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2),E3)),C3)),D3) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3375_real__of__int__div4,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,Xb))),divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),Xb))) ).

% real_of_int_div4
tff(fact_3376_real__of__int__div,axiom,
    ! [D3: int,Nb: int] :
      ( dvd_dvd(int,D3,Nb)
     => ( aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,D3)) = divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),D3)) ) ) ).

% real_of_int_div
tff(fact_3377_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_3378_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) )
         => ( 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_ii(A,fun(A,fun(nat,fun(nat,A))),Xb),Y),Nb)),set_ord_atMost(nat,Nb)) ) ) ) ).

% exp_series_add_commuting
tff(fact_3379_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : exp(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_ij(A,fun(nat,A),Xb))) ) ).

% exp_first_term
tff(fact_3380_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_3381_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_3382_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_3383_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_3384_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_3385_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_3386_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_3387_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_3388_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_3389_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)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_3390_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ) ).

% of_int_leD
tff(fact_3391_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)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_pos
tff(fact_3392_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,B2)),zero_zero(A))
        <=> ( ( aa(real,$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_3393_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),real_V8093663219630862766scaleR(A,A2,B2))
        <=> ( ( aa(real,$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_3394_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xb) ) ) ) ).

% of_int_lessD
tff(fact_3395_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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3396_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C3: A,D3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),real_V8093663219630862766scaleR(A,A2,C3)),real_V8093663219630862766scaleR(A,B2,D3)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3397_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),real_V8093663219630862766scaleR(A,A2,Xb)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_3398_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)),real_V8093663219630862766scaleR(A,A2,B2)) ) ) ).

% split_scaleR_pos_le
tff(fact_3399_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)),real_V8093663219630862766scaleR(A,A2,Xb)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_3400_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),real_V8093663219630862766scaleR(A,A2,Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_3401_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),real_V8093663219630862766scaleR(A,A2,Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_3402_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)),real_V8093663219630862766scaleR(A,A2,B2)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_3403_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),real_V8093663219630862766scaleR(A,A2,Xb)),Xb) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_3404_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int)))) ) ) ).

% floor_exists
tff(fact_3405_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [X4: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X4)),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),one_one(int))))
          & ! [Y4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y4)),Xb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y4),one_one(int)))) )
             => ( Y4 = X4 ) ) ) ) ).

% floor_exists1
tff(fact_3406_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2)) ) ).

% of_int_neg_numeral
tff(fact_3407_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: 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_3408_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$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_3409_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E3: real] :
      ( ! [D2: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D2),E2)
         => ( aa(real,$o,P,D2)
           => 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)),E3)
         => aa(real,$o,P,E3) ) ) ) ).

% forall_pos_mono_1
tff(fact_3410_real__arch__inverse,axiom,
    ! [E3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
    <=> ? [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))),E3) ) ) ).

% real_arch_inverse
tff(fact_3411_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E3: real] :
      ( ! [D2: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D2),E2)
         => ( aa(real,$o,P,D2)
           => 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)),E3)
         => aa(real,$o,P,E3) ) ) ) ).

% forall_pos_mono
tff(fact_3412_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),aa(int,real,ring_1_of_int(real),Nb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Mb)),one_one(real))) ) ).

% int_le_real_less
tff(fact_3413_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),aa(int,real,ring_1_of_int(real),Nb)),one_one(real))),aa(int,real,ring_1_of_int(real),Mb)) ) ).

% int_less_real_le
tff(fact_3414_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I3: int] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),pi) ) ).

% sin_zero_iff_int2
tff(fact_3415_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,K2: nat] : exp(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ig(A,fun(nat,A),Xb)),set_ord_lessThan(nat,K2))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ik(A,fun(nat,fun(nat,A)),Xb),K2))) ) ).

% exp_first_terms
tff(fact_3416_real__of__int__div__aux,axiom,
    ! [Xb: int,D3: int] : divide_divide(real,aa(int,real,ring_1_of_int(real),Xb),aa(int,real,ring_1_of_int(real),D3)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),divide_divide(int,Xb,D3))),divide_divide(real,aa(int,real,ring_1_of_int(real),modulo_modulo(int,Xb,D3)),aa(int,real,ring_1_of_int(real),D3))) ).

% real_of_int_div_aux
tff(fact_3417_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_il(A,fun(nat,A),Xb)) ) ).

% summable_exp
tff(fact_3418_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_3419_odd__numeral__BitM,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [W: num] : ~ 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_3420_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_3421_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_3422_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_im(A,fun(nat,A),Xb),sin(A,Xb)) ) ).

% sin_converges
tff(fact_3423_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sin(A,X) = suminf(A,aTP_Lamp_im(A,fun(nat,A),X)) ) ).

% sin_def
tff(fact_3424_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_in(A,fun(nat,A),Xb),cos(A,Xb)) ) ).

% cos_converges
tff(fact_3425_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : cos(A,X) = suminf(A,aTP_Lamp_in(A,fun(nat,A),X)) ) ).

% cos_def
tff(fact_3426_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_io(A,fun(nat,real),Xb)) ) ).

% summable_norm_sin
tff(fact_3427_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_ip(A,fun(nat,real),Xb)) ) ).

% summable_norm_cos
tff(fact_3428_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: 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_iq(A,fun(nat,A),Xb))) ) ).

% exp_first_two_terms
tff(fact_3429_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),divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),Xb))),aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,Xb)))) ).

% real_of_int_div2
tff(fact_3430_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),divide_divide(real,aa(int,real,ring_1_of_int(real),Nb),aa(int,real,ring_1_of_int(real),Xb))),aa(int,real,ring_1_of_int(real),divide_divide(int,Nb,Xb)))),one_one(real)) ).

% real_of_int_div3
tff(fact_3431_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_ir(A,fun(nat,A),Xb),sin(A,Xb)) ) ).

% sin_minus_converges
tff(fact_3432_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_is(A,fun(nat,A),Xb),cos(A,Xb)) ) ).

% cos_minus_converges
tff(fact_3433_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K2: int] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(int,A,ring_1_of_int(A),K2))
        <=> dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2) ) ) ).

% even_of_int_iff
tff(fact_3434_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),exp(real,Xb)),aa(real,real,inverse_inverse(real),exp(real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_3435_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_3436_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_3437_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = one_one(real) )
    <=> ? [X3: int] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),X3)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3438_tan__cot,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),minus_minus(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_3439_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),divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),exp(real,Xb)),aa(real,real,inverse_inverse(real),exp(real,Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ).

% real_le_x_sinh
tff(fact_3440_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),divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),exp(real,Xb)),aa(real,real,inverse_inverse(real),exp(real,Xb))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% real_le_abs_sinh
tff(fact_3441_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_3442_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K: 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),aa(int,real,ring_1_of_int(real),K)),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_3443_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = zero_zero(real) )
    <=> ? [I3: int] :
          ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),I3)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_3444_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I3: int] :
          ( dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),I3)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I3)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_3445_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_iu(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_3446_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K2: int] :
          aa(int,A,ring_1_of_int(A),K2) = $ite(
            K2 = zero_zero(int),
            zero_zero(A),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)),
              aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K2))),
              $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))),aa(int,A,ring_1_of_int(A),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),
                $ite(modulo_modulo(int,K2,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_3447_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),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_3448_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),aa(int,A,ring_1_of_int(A),Nb)))),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_3449_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),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))),Xb))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% of_int_round_abs_le
tff(fact_3450_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_iv(A,fun(nat,A),Xb),sinh(A,Xb)) ) ).

% sinh_converges
tff(fact_3451_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_3452_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_3453_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_3454_divide__complex__def,axiom,
    ! [Xb: complex,Y: 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_3455_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_3456_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),aa(int,A,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),aa(int,A,ring_1_of_int(A),Mb)))) ) ).

% round_diff_minimal
tff(fact_3457_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,inverse_inverse(complex),complex2(A2,B2)) = complex2(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))))),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_3458_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : sinh(A,Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Z)),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_3459_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sinh(A,Xb) = 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),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% sinh_def
tff(fact_3460_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)) = 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_3461_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% of_int_round_le
tff(fact_3462_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),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))) ) ).

% of_int_round_ge
tff(fact_3463_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),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))) ) ).

% of_int_round_gt
tff(fact_3464_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sums(A,aTP_Lamp_iw(A,fun(nat,A),Xb),cosh(A,Xb)) ) ).

% cosh_converges
tff(fact_3465_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(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_3466_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_3467_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),divide_divide(real,aa(real,real,ln_ln(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_3468_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_3469_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_3470_log__one,axiom,
    ! [A2: real] : aa(real,real,log(A2),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_3471_norm__ii,axiom,
    real_V7770717601297561774m_norm(complex,imaginary_unit) = one_one(real) ).

% norm_ii
tff(fact_3472_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_3473_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_3474_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_3475_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_3476_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_3477_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_3478_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_3479_divide__i,axiom,
    ! [Xb: 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_3480_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : divide_divide(complex,Z,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),Nb)),imaginary_unit)) = 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_3481_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_3482_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_3483_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_3484_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_3485_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_3486_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_3487_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_3488_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_3489_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_3490_complex__i__not__one,axiom,
    imaginary_unit != one_one(complex) ).

% complex_i_not_one
tff(fact_3491_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_3492_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_3493_log__ln,axiom,
    ! [Xb: real] : aa(real,real,ln_ln(real),Xb) = aa(real,real,log(exp(real,one_one(real))),Xb) ).

% log_ln
tff(fact_3494_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_3495_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_3496_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_3497_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_3498_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_3499_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_3500_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)) = exp(A,Xb) ) ).

% sinh_plus_cosh
tff(fact_3501_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)) = exp(A,Xb) ) ).

% cosh_plus_sinh
tff(fact_3502_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tanh(A),Xb) = divide_divide(A,sinh(A,Xb),cosh(A,Xb)) ) ).

% tanh_def
tff(fact_3503_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_3504_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_3505_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) = divide_divide(real,aa(real,real,log(A2),Xb),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_3506_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_3507_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_3508_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)) = exp(A,aa(A,A,uminus_uminus(A),Xb)) ) ).

% cosh_minus_sinh
tff(fact_3509_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),exp(A,aa(A,A,uminus_uminus(A),Xb))) ) ).

% sinh_minus_cosh
tff(fact_3510_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_3511_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),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_3512_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_3513_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_3514_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_3515_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,cot(A),X) = divide_divide(A,cos(A,X),sin(A,X)) ) ).

% cot_def
tff(fact_3516_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_3517_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_3518_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_3519_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),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_3520_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_3521_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)) = 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_3522_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_3523_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_3524_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : cosh(A,Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,Z)),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_3525_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_3526_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_3527_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_3528_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_3529_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),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_3530_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_3531_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : cosh(A,Xb) = 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),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% cosh_def
tff(fact_3532_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)) = 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_3533_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),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_3534_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_3535_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))))),exp(real,one_one(real)))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq2
tff(fact_3536_tan__cot_H,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),minus_minus(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_3537_Arg__minus__ii,axiom,
    arg(aa(complex,complex,uminus_uminus(complex),imaginary_unit)) = 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_3538_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K2: 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)),K2)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K2))) = 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)),K2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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_3539_Arg__ii,axiom,
    arg(imaginary_unit) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% Arg_ii
tff(fact_3540_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K2: 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)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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),K2))) = 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_3541_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),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_3542_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb)) = Xb )
        <=> ? [N4: int] : Xb = aa(int,A,ring_1_of_int(A),N4) ) ) ).

% of_int_ceiling_cancel
tff(fact_3543_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_3544_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_3545_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),aa(int,A,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_3546_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),aa(int,A,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_3547_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_3548_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_3549_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_3550_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_3551_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_3552_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_3553_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_3554_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_3555_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_3556_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_3557_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_3558_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_3559_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_3560_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_3561_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_3562_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_3563_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,uminus_uminus(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_3564_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_3565_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_3566_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_3567_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_3568_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,uminus_uminus(real),divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,uminus_uminus(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_3569_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_3570_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_3571_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_3572_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))) ) ).

% le_of_int_ceiling
tff(fact_3573_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_3574_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),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_3575_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),A2) ) ) ).

% ceiling_le
tff(fact_3576_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),aa(int,A,ring_1_of_int(A),Z)),Xb) ) ) ).

% less_ceiling_iff
tff(fact_3577_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_3578_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_3579_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),one_one(A))),R2) ) ).

% of_int_ceiling_diff_one_le
tff(fact_3580_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: int,B2: int] : archimedean_ceiling(A,divide_divide(A,aa(int,A,ring_1_of_int(A),A2),aa(int,A,ring_1_of_int(A),B2))) = aa(int,int,uminus_uminus(int),divide_divide(int,aa(int,int,uminus_uminus(int),A2),B2)) ) ).

% ceiling_divide_eq_div
tff(fact_3581_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))),one_one(A))),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))) ) ) ).

% ceiling_correct
tff(fact_3582_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xb) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_3583_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( archimedean_ceiling(A,Xb) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_3584_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Ta))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),aa(int,A,ring_1_of_int(A),I3)) )
             => aa(int,$o,P,I3) ) ) ) ).

% ceiling_split
tff(fact_3585_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_3586_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),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_3587_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),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb) ) ) ).

% le_ceiling_iff
tff(fact_3588_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P2,Q3)))),Q3)) ) ) ).

% ceiling_divide_upper
tff(fact_3589_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$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),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P2,Q3)))),one_one(A))),Q3)),P2) ) ) ).

% ceiling_divide_lower
tff(fact_3590_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Nb)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,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_3591_cis__minus__pi__half,axiom,
    cis(aa(real,real,uminus_uminus(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_3592_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K2: 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),K2)),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),K2))),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),K2),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_3593_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K2: 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)),K2)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K2))) = 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)),K2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),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_3594_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K2: 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)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),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),K2))) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_3595_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_ix(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_3596_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_3597_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb)) = Xb )
        <=> ? [N4: int] : Xb = aa(int,A,ring_1_of_int(A),N4) ) ) ).

% of_int_floor_cancel
tff(fact_3598_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_3599_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_3600_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_3601_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_3602_norm__cis,axiom,
    ! [A2: real] : real_V7770717601297561774m_norm(complex,cis(A2)) = one_one(real) ).

% norm_cis
tff(fact_3603_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_3604_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_3605_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_3606_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_3607_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_3608_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_3609_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),aa(int,A,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_3610_cis__pi,axiom,
    cis(pi) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% cis_pi
tff(fact_3611_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_3612_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_3613_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_3614_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_3615_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_3616_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_3617_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_3618_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_3619_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_3620_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_3621_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_3622_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_3623_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_3624_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_3625_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2))) = divide_divide(int,aa(num,int,numeral_numeral(int),A2),aa(num,int,numeral_numeral(int),B2)) ).

% floor_divide_eq_div_numeral
tff(fact_3626_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_3627_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_3628_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_3629_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_3630_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_3631_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_3632_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_3633_cis__pi__half,axiom,
    cis(divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = imaginary_unit ).

% cis_pi_half
tff(fact_3634_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),B2))) = divide_divide(int,one_one(int),aa(num,int,numeral_numeral(int),B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_3635_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_3636_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),divide_divide(real,aa(num,real,numeral_numeral(real),A2),aa(num,real,numeral_numeral(real),B2)))) = 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_3637_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_3638_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_3639_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))),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_3640_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),B2)))) = 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_3641_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_3642_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_3643_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb))),Xb) ) ).

% of_int_floor_le
tff(fact_3644_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_3645_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_3646_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_3647_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_3648_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_3649_cis__divide,axiom,
    ! [A2: real,B2: real] : 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_3650_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),aa(int,A,ring_1_of_int(A),Z)),Xb) ) ) ).

% le_floor_iff
tff(fact_3651_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),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_3652_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_3653_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_3654_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_3655_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),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_3656_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),aa(int,A,ring_1_of_int(A),Z)),Xb)) ) ).

% int_add_floor
tff(fact_3657_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_3658_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_3659_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_3660_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_3661_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K2: int,L: int] : archim6421214686448440834_floor(A,divide_divide(A,aa(int,A,ring_1_of_int(A),K2),aa(int,A,ring_1_of_int(A),L))) = divide_divide(int,K2,L) ) ).

% floor_divide_of_int_eq
tff(fact_3662_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Nb: nat] :
          ( ( Xb = aa(int,A,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_3663_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C3: real] : divide_divide(real,A2,powr(real,B2,C3)) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),powr(real,B2,aa(real,real,uminus_uminus(real),C3))) ).

% divide_powr_uminus
tff(fact_3664_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_3665_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_3666_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_3667_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)) = divide_divide(A,powr(A,W,Z1),powr(A,W,Z22)) ) ).

% powr_diff
tff(fact_3668_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_3669_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K2: 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)) = K2 )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,aa(int,real,ring_1_of_int(real),K2))),Xb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),powr(real,B2,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_3670_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Mb: nat,Nb: nat] : archim6421214686448440834_floor(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),divide_divide(nat,Mb,Nb)) ) ).

% floor_divide_of_nat_eq
tff(fact_3671_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_3672_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_3673_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_3674_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_3675_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_ceiling(A,Xb) = $ite(Xb = aa(int,A,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_3676_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_3677_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_3678_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,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),aa(int,real,ring_1_of_int(real),Nb)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_3679_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_3680_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_3681_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_3682_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_3683_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,Ta))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I3)),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))) )
             => aa(int,$o,P,I3) ) ) ) ).

% floor_split
tff(fact_3684_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,Xb) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_3685_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
           => ( archim6421214686448440834_floor(A,Xb) = Z ) ) ) ) ).

% floor_unique
tff(fact_3686_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)) = divide_divide(A,one_one(A),powr(A,Xb,A2)) ) ).

% powr_minus_divide
tff(fact_3687_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_3688_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),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb) ) ) ).

% less_floor_iff
tff(fact_3689_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),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_3690_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb))),Xb)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(int,A,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_3691_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))) = divide_divide(real,one_one(real),Xb) ) ) ).

% powr_neg_one
tff(fact_3692_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_3693_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_3694_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_3695_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_3696_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_3697_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,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),aa(int,real,ring_1_of_int(real),Nb)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_3698_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,divide_divide(real,A2,aa(int,real,ring_1_of_int(real),B2))) = divide_divide(int,archim6421214686448440834_floor(real,A2),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_3699_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P2,Q3)))),Q3)),P2) ) ) ).

% floor_divide_lower
tff(fact_3700_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)),divide_divide(real,powr(real,Xb,A2),A2)) ) ) ).

% ln_powr_bound
tff(fact_3701_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_3702_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_3703_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_3704_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),divide_divide(real,powr(real,B2,Y),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_3705_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A,A2: A] :
          powr(A,Xb,A2) = $ite(Xb = zero_zero(A),zero_zero(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_3706_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$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),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P2,Q3)))),one_one(A))),Q3)) ) ) ).

% floor_divide_upper
tff(fact_3707_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),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% round_def
tff(fact_3708_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_3709_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,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_3710_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))) = 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_3711_pi__half,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) = the(real,aTP_Lamp_iy(real,$o)) ).

% pi_half
tff(fact_3712_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_iy(real,$o))) ).

% pi_def
tff(fact_3713_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),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_3714_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_iz(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_3715_old_Orec__prod__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: fun(A,fun(B,C)),Xa: product_prod(A,B)] : product_rec_prod(A,B,C,X,Xa) = the(C,product_rec_set_prod(A,B,C,X,Xa)) ).

% old.rec_prod_def
tff(fact_3716_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_ja(nat,fun(nat,complex),Nb),set_ord_lessThan(nat,Nb),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_bu(nat,fun(complex,$o),Nb))) ) ).

% bij_betw_roots_unity
tff(fact_3717_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)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))) ) ).

% arcosh_def
tff(fact_3718_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),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_3719_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,$o)),aTP_Lamp_jb(A,fun(B,fun(A,fun(B,$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_3720_of__real__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( real_Vector_of_real(A,one_one(real)) = one_one(A) ) ) ).

% of_real_1
tff(fact_3721_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real] :
          ( ( real_Vector_of_real(A,Xb) = one_one(A) )
        <=> ( Xb = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_3722_of__real__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : 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),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ).

% of_real_mult
tff(fact_3723_of__real__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : real_Vector_of_real(A,aa(num,real,numeral_numeral(real),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_real_numeral
tff(fact_3724_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: real,Y: real] : real_Vector_of_real(A,divide_divide(real,Xb,Y)) = divide_divide(A,real_Vector_of_real(A,Xb),real_Vector_of_real(A,Y)) ) ).

% of_real_divide
tff(fact_3725_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Nb: nat] : 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),real_Vector_of_real(A,Xb)),Nb) ) ).

% of_real_power
tff(fact_3726_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : 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),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ).

% of_real_add
tff(fact_3727_of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : 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),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ).

% of_real_diff
tff(fact_3728_of__real__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : 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_3729_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,real_Vector_of_real(A,pi)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% cos_of_real_pi
tff(fact_3730_exp__pi__i_H,axiom,
    exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,pi))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i'
tff(fact_3731_exp__pi__i,axiom,
    exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,pi)),imaginary_unit)) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i
tff(fact_3732_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),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_3733_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),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_3734_exp__two__pi__i_H,axiom,
    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),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_3735_exp__two__pi__i,axiom,
    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))),real_Vector_of_real(complex,pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_3736_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3737_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = one_one(A) ) ) ).

% sin_of_real_pi_half
tff(fact_3738_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A4: complex,R4: real] : Z = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R4)),exp(complex,A4)) ).

% complex_exp_exists
tff(fact_3739_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R2: real,Xb: A] : real_V8093663219630862766scaleR(A,R2,Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,R2)),Xb) ) ).

% scaleR_conv_of_real
tff(fact_3740_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R2: real] : real_Vector_of_real(A,R2) = real_V8093663219630862766scaleR(A,R2,one_one(A)) ) ).

% of_real_def
tff(fact_3741_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_3742_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_3743_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_3744_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,Xb: real] :
          ( ( Y != zero_zero(real) )
         => ( real_Vector_of_real(A,divide_divide(real,Xb,Y)) = divide_divide(A,real_Vector_of_real(A,Xb),real_Vector_of_real(A,Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_3745_Complex__mult__complex__of__real,axiom,
    ! [Xb: real,Y: real,R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(Xb,Y)),real_Vector_of_real(complex,R2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),Xb),R2),aa(real,real,aa(real,fun(real,real),times_times(real),Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_3746_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xb: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R2)),complex2(Xb,Y)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_3747_cis__conv__exp,axiom,
    ! [B2: real] : cis(B2) = exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,B2))) ).

% cis_conv_exp
tff(fact_3748_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),real_Vector_of_real(A,real_V7770717601297561774m_norm(A,Xb))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_3749_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S4: set(A),T3: set(B),H: fun(A,B),S3: set(A),T2: set(B),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S4)
         => ( aa(set(B),$o,finite_finite(B),T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
             => ( ! [A4: A] :
                    ( member(A,A4,S4)
                   => ( aa(B,C,G3,aa(A,B,H,A4)) = zero_zero(C) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T3)
                     => ( aa(B,C,G3,B4) = zero_zero(C) ) )
                 => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_jc(fun(A,B),fun(fun(B,C),fun(A,C)),H),G3)),S3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G3),T2) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_3750_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S4: set(A),T3: set(B),H: fun(A,B),S3: set(A),T2: set(B),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S4)
         => ( aa(set(B),$o,finite_finite(B),T3)
           => ( bij_betw(A,B,H,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),S4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T2),T3))
             => ( ! [A4: A] :
                    ( member(A,A4,S4)
                   => ( aa(B,C,G3,aa(A,B,H,A4)) = one_one(C) ) )
               => ( ! [B4: B] :
                      ( member(B,B4,T3)
                     => ( aa(B,C,G3,B4) = 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_jd(fun(A,B),fun(fun(B,C),fun(A,C)),H),G3)),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G3),T2) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_3751_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),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb))) ) ).

% frac_def
tff(fact_3752_i__complex__of__real,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,R2)) = complex2(zero_zero(real),R2) ).

% i_complex_of_real
tff(fact_3753_complex__of__real__i,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R2)),imaginary_unit) = complex2(zero_zero(real),R2) ).

% complex_of_real_i
tff(fact_3754_Complex__eq,axiom,
    ! [A2: real,B2: real] : complex2(A2,B2) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,A2)),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,B2))) ).

% Complex_eq
tff(fact_3755_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),real_Vector_of_real(A,B2)),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_3756_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_3757_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_3758_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R4: real,A4: real] : Z = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R4)),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,cos(real,A4))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A4))))) ).

% complex_split_polar
tff(fact_3759_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),aa(int,A,ring_1_of_int(A),Mb)),real_Vector_of_real(A,Xb))) = real_Vector_of_real(A,cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),Mb)),Xb))) ) ).

% cos_int_times_real
tff(fact_3760_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),aa(int,A,ring_1_of_int(A),Mb)),real_Vector_of_real(A,Xb))) = real_Vector_of_real(A,sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),Mb)),Xb))) ) ).

% sin_int_times_real
tff(fact_3761_floor__real__def,axiom,
    ! [Xb: real] : archim6421214686448440834_floor(real,Xb) = the(int,aTP_Lamp_je(real,fun(int,$o),Xb)) ).

% floor_real_def
tff(fact_3762_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),divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Xb)) ) ).

% cos_sin_eq
tff(fact_3763_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),divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Xb)) ) ).

% sin_cos_eq
tff(fact_3764_cmod__unit__one,axiom,
    ! [A2: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,cos(real,A2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A2))))) = one_one(real) ).

% cmod_unit_one
tff(fact_3765_cmod__complex__polar,axiom,
    ! [R2: real,A2: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R2)),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,cos(real,A2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A2)))))) = aa(real,real,abs_abs(real),R2) ).

% cmod_complex_polar
tff(fact_3766_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_3767_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),divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% minus_sin_cos_eq
tff(fact_3768_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)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))) ) ).

% arsinh_def
tff(fact_3769_csqrt__ii,axiom,
    csqrt(imaginary_unit) = divide_divide(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),imaginary_unit),real_Vector_of_real(complex,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% csqrt_ii
tff(fact_3770_modulo__int__unfold,axiom,
    ! [K2: 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),K2)),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),K2) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),aa(nat,int,semiring_1_of_nat(int),Mb)),
        $ite(aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Mb,Nb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,Nb,Mb))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,Mb,Nb))))) ) ).

% modulo_int_unfold
tff(fact_3771_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_3772_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_3773_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A2)) = aa(A,A,sgn_sgn(A),A2) ) ).

% sgn_sgn
tff(fact_3774_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_3775_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_3776_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_3777_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_3778_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,sgn_sgn(A),divide_divide(A,A2,B2)) = divide_divide(A,aa(A,A,sgn_sgn(A),A2),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_divide
tff(fact_3779_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_3780_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_3781_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% csqrt_eq_1
tff(fact_3782_csqrt__1,axiom,
    csqrt(one_one(complex)) = one_one(complex) ).

% csqrt_1
tff(fact_3783_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_3784_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_3785_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: 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_3786_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_3787_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_3788_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_3789_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_3790_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_3791_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_3792_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_3793_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_3794_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R2: int] :
      ( dvd_dvd(int,L,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(int,int,sgn_sgn(int),R2)))
    <=> ( dvd_dvd(int,L,K2)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_3795_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( dvd_dvd(int,L,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R2)),K2))
    <=> ( dvd_dvd(int,L,K2)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_3796_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K2: int] :
      ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R2)),K2)
    <=> ( dvd_dvd(int,L,K2)
        & ( ( R2 = zero_zero(int) )
         => ( K2 = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_3797_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K2: int] :
      ( dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R2)),L),K2)
    <=> ( dvd_dvd(int,L,K2)
        & ( ( R2 = zero_zero(int) )
         => ( K2 = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_3798_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_3799_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_3800_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_3801_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(int,A,ring_1_of_int(A),bit_se2239418461657761734s_mask(int,Nb)) = bit_se2239418461657761734s_mask(A,Nb) ) ).

% of_int_mask_eq
tff(fact_3802_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_3803_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_3804_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_3805_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_3806_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_3807_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_3808_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_3809_sgn__not__eq__imp,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),A2) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_3810_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_3811_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K2: A] : aa(A,A,abs_abs(A),K2) = aa(A,A,aa(A,fun(A,A),times_times(A),K2),aa(A,A,sgn_sgn(A),K2)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_3812_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_3813_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_3814_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_3815_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N: nat,L3: int] : K2 != 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_3816_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_3817_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_3818_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_3819_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_3820_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_3821_sgn__mod,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero(int) )
     => ( ~ dvd_dvd(int,L,K2)
       => ( aa(int,int,sgn_sgn(int),modulo_modulo(int,K2,L)) = aa(int,int,sgn_sgn(int),L) ) ) ) ).

% sgn_mod
tff(fact_3822_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_3823_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_3824_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_3825_zsgn__def,axiom,
    ! [I: int] :
      aa(int,int,sgn_sgn(int),I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_3826_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_3827_div__sgn__abs__cancel,axiom,
    ! [V: int,K2: int,L: int] :
      ( ( V != zero_zero(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),K2)),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))) = divide_divide(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_3828_div__dvd__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( dvd_dvd(int,L,K2)
     => ( divide_divide(int,K2,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),K2)),aa(int,int,sgn_sgn(int),L))),divide_divide(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_3829_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_3830_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_3831_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_3832_div__noneq__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K2) != aa(int,int,sgn_sgn(int),L) )
       => ( divide_divide(int,K2,L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),divide_divide(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L)))),aa($o,int,zero_neq_one_of_bool(int),~ dvd_dvd(int,L,K2))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_3833_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K2: int,Q3: int] :
      ( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),L))
       => ( ( K2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R2) )
         => eucl_rel_int(K2,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_3834_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( 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_3835_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_3836_mask__half__int,axiom,
    ! [Nb: nat] : 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_3837_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_3838_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_3839_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) ) )
       => ( ! [Q2: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q2),A22) ) ) )
         => ~ ! [R4: int,Q2: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R4) )
               => ( ( aa(int,int,sgn_sgn(int),R4) = aa(int,int,sgn_sgn(int),A22) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R4)),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),Q2),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_3840_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_3841_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : size_num(aa(num,num,bit0,X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_3842_divide__int__unfold,axiom,
    ! [K2: int,Mb: nat,L: int,Nb: nat] :
      divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K2)),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),K2) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Mb,Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,Mb,Nb)),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,Nb,Mb)))))) ) ).

% divide_int_unfold
tff(fact_3843_floor__rat__def,axiom,
    ! [Xb: rat] : archim6421214686448440834_floor(rat,Xb) = the(int,aTP_Lamp_jf(rat,fun(int,$o),Xb)) ).

% floor_rat_def
tff(fact_3844_bij__betw__nth__root__unity,axiom,
    ! [C3: complex,Nb: nat] :
      ( ( C3 != 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),real_Vector_of_real(complex,aa(real,real,root(Nb),real_V7770717601297561774m_norm(complex,C3)))),cis(divide_divide(real,arg(C3),aa(nat,real,semiring_1_of_nat(real),Nb))))),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_bu(nat,fun(complex,$o),Nb)),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_jg(complex,fun(nat,fun(complex,$o)),C3),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_3845_divide__int__def,axiom,
    ! [K2: int,L: int] :
      divide_divide(int,K2,L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(int,L,K2)))))) ) ).

% divide_int_def
tff(fact_3846_powr__int,axiom,
    ! [Xb: real,I: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(int,nat,nat2,I)),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),I))))) ) ) ).

% powr_int
tff(fact_3847_nat__numeral,axiom,
    ! [K2: num] : aa(int,nat,nat2,aa(num,int,numeral_numeral(int),K2)) = aa(num,nat,numeral_numeral(nat),K2) ).

% nat_numeral
tff(fact_3848_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_3849_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_3850_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_3851_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_3852_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_3853_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_3854_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_3855_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_3856_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_3857_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_3858_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_3859_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_3860_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_3861_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_3862_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_3863_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_3864_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_3865_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_3866_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_3867_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_3868_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_3869_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_3870_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_3871_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_3872_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_3873_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_3874_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_3875_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_3876_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_3877_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_3878_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_3879_sgn__rat__def,axiom,
    ! [A2: rat] :
      aa(rat,rat,sgn_sgn(rat),A2) = $ite(
        A2 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A2),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_3880_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
     => ~ ! [S: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),S)
           => ! [T4: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T4)
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_3881_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_3882_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_3883_nat__numeral__as__int,axiom,
    ! [X: num] : aa(num,nat,numeral_numeral(nat),X) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X)) ).

% nat_numeral_as_int
tff(fact_3884_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_3885_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_3886_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_3887_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_3888_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_3889_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,$o),ord_less(nat),zero_zero(nat)),Nb)
         => ! [Y5: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y5)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y5)),Nb)) = Xb )
             => aa(real,$o,P,Y5) ) ) ) ) ).

% split_root
tff(fact_3890_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_3891_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_3892_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K2: 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),K2)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),K2) ) ) ).

% real_root_power
tff(fact_3893_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_3894_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_3895_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R2)))) ) ).

% of_nat_ceiling
tff(fact_3896_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_3897_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_3898_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_3899_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_3900_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_3901_nat__plus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_3902_nat__times__as__int,axiom,
    ! [X: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_3903_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_3904_nat__minus__as__int,axiom,
    ! [X: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_minus_as_int
tff(fact_3905_nat__div__as__int,axiom,
    ! [X: nat,Xa: nat] : divide_divide(nat,X,Xa) = aa(int,nat,nat2,divide_divide(int,aa(nat,int,semiring_1_of_nat(int),X),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_3906_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_3907_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N3: 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),N3)
       => ( 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(N3),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_3908_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% sqrt_def
tff(fact_3909_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_3910_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_3911_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
         => aa(A,$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,R2)))),R2) ) ) ).

% of_nat_floor
tff(fact_3912_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_3913_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_3914_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_3915_le__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(int,nat,nat2,K2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K2) ) ) ).

% le_nat_iff
tff(fact_3916_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_3917_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_3918_Suc__as__int,axiom,
    ! [X: nat] : aa(nat,nat,suc,X) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X)),one_one(int))) ).

% Suc_as_int
tff(fact_3919_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_3920_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_3921_nat__abs__triangle__ineq,axiom,
    ! [K2: 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),K2),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K2))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_3922_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,divide_divide(int,Xb,Y)) = divide_divide(nat,aa(int,nat,nat2,Xb),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_3923_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,divide_divide(int,Xb,Y)) = divide_divide(nat,aa(int,nat,nat2,Xb),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_3924_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_3925_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_3926_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_3927_div__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] : divide_divide(int,aa(int,int,abs_abs(int),K2),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_3928_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_3929_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_3930_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_3931_real__root__strict__increasing,axiom,
    ! [Nb: nat,N3: 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),N3)
       => ( 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(N3),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_3932_real__root__decreasing,axiom,
    ! [Nb: nat,N3: 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),N3)
       => ( 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(N3),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_3933_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_3934_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_3935_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_3936_odd__real__root__pow,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ 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_3937_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( ~ 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_3938_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ 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_3939_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_3940_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_3941_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_3942_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_3943_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_3944_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_3945_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_3946_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_3947_real__root__increasing,axiom,
    ! [Nb: nat,N3: 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),N3)
       => ( 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(N3),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_3948_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)) = divide_divide(real,aa(real,real,ln_ln(real),B2),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% ln_root
tff(fact_3949_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)) = divide_divide(real,aa(real,real,log(B2),A2),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_root
tff(fact_3950_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_3951_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,divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_3952_even__nat__iff,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(int,nat,nat2,K2))
      <=> dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2) ) ) ).

% even_nat_iff
tff(fact_3953_modulo__int__def,axiom,
    ! [K2: int,L: int] :
      modulo_modulo(int,K2,L) = $ite(
        L = zero_zero(int),
        K2,
        $ite(aa(int,int,sgn_sgn(int),K2) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($o,int,zero_neq_one_of_bool(int),~ dvd_dvd(int,L,K2)))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))))) ) ).

% modulo_int_def
tff(fact_3954_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero(real) )
     => ( aa(real,real,arctan,divide_divide(real,one_one(real),Xb)) = aa(real,real,aa(real,fun(real,real),minus_minus(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_3955_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))),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_3956_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6)
     => ( aa(set(A),$o,finite_finite(A),X6)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,count_list(A,Xs)),X6) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_3957_and__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = $ite(
        ( ( K2 = zero_zero(int) )
        | ( L = zero_zero(int) ) ),
        zero_zero(int),
        $ite(
          K2 = aa(int,int,uminus_uminus(int),one_one(int)),
          L,
          $ite(L = aa(int,int,uminus_uminus(int),one_one(int)),K2,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,K2,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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% and_int_unfold
tff(fact_3958_power__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K2: num,L: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),K2)),aa(num,nat,numeral_numeral(nat),L)) = aa(num,A,numeral_numeral(A),pow(K2,L)) ) ).

% power_numeral
tff(fact_3959_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_3960_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_3961_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_3962_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_3963_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_3964_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_3965_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_3966_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_3967_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_3968_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_concat_bit(Nb,K2),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) ).

% concat_bit_of_zero_2
tff(fact_3969_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_3970_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_3971_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_3972_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_3973_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_3974_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_3975_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,K2: int] : aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_3976_and__nonnegative__int__iff,axiom,
    ! [K2: 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),K2),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_3977_and__negative__int__iff,axiom,
    ! [K2: 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),K2),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_3978_count__notin,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ 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_3979_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_3980_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_3981_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_3982_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_3983_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_3984_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_3985_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_3986_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_3987_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_3988_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_3989_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_3990_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_3991_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_3992_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( 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) )
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_3993_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_3994_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_3995_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_3996_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_3997_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_3998_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_3999_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_4000_diff__rat__def,axiom,
    ! [Q3: rat,R2: rat] : aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q3),R2) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q3),aa(rat,rat,uminus_uminus(rat),R2)) ).

% diff_rat_def
tff(fact_4001_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q3: 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),Q3)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_4002_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_4003_nat__take__bit__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(int,nat,nat2,K2)) ) ) ).

% nat_take_bit_eq
tff(fact_4004_take__bit__nat__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),aa(int,nat,nat2,K2)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)) ) ) ).

% take_bit_nat_eq
tff(fact_4005_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_4006_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K2: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(int,A,ring_1_of_int(A),K2)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)) ) ).

% take_bit_of_int
tff(fact_4007_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K2: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(int,A,ring_1_of_int(A),K2)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_and_eq
tff(fact_4008_take__bit__diff,axiom,
    ! [Nb: nat,K2: 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),K2)),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),K2),L)) ).

% take_bit_diff
tff(fact_4009_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = 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),C3)) ) ).

% and.assoc
tff(fact_4010_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_4011_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C3: 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),C3)) = 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),C3)) ) ).

% and.left_commute
tff(fact_4012_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_4013_take__bit__mult,axiom,
    ! [Nb: nat,K2: 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),K2)),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),K2),L)) ).

% take_bit_mult
tff(fact_4014_take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),K2)) ).

% take_bit_minus
tff(fact_4015_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_4016_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_4017_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_4018_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K2: int,L: int,R2: int,S2: int] :
      ( ( aa(int,int,bit_concat_bit(Nb,K2),L) = aa(int,int,bit_concat_bit(Nb,R2),S2) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),R2) )
        & ( L = S2 ) ) ) ).

% concat_bit_eq_iff
tff(fact_4019_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_4020_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K2: 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),K2)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_4021_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_4022_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_4023_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_4024_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_4025_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_4026_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),K2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_4027_take__bit__nonnegative,axiom,
    ! [Nb: nat,K2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)) ).

% take_bit_nonnegative
tff(fact_4028_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_4029_take__bit__int__greater__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_4030_not__take__bit__negative,axiom,
    ! [Nb: nat,K2: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_4031_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_4032_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_4033_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_4034_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_4035_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : pow(Xb,one2) = Xb ).

% pow.simps(1)
tff(fact_4036_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_4037_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_4038_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_4039_and__less__eq,axiom,
    ! [L: int,K2: 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),K2),L)),K2) ) ).

% and_less_eq
tff(fact_4040_take__bit__decr__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),one_one(int))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_4041_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( 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))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_and_iff
tff(fact_4042_even__and__iff__int,axiom,
    ! [K2: int,L: int] :
      ( 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),K2),L))
    <=> ( dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2)
        | dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L) ) ) ).

% even_and_iff_int
tff(fact_4043_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_4044_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_4045_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K2))) = 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),K2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_bit0
tff(fact_4046_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_4047_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_4048_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_4049_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_4050_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_4051_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_4052_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_4053_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),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_4054_take__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) = modulo_modulo(int,K2,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_4055_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) )
        <=> 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_4056_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K2: 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,K2))) = 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),K2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_numeral_bit0
tff(fact_4057_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_4058_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: 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,K2)))) = 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),K2)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_4059_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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_4060_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),K2)
    <=> 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)),K2) ) ).

% take_bit_int_less_self_iff
tff(fact_4061_take__bit__int__eq__self,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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),K2) = K2 ) ) ) ).

% take_bit_int_eq_self
tff(fact_4062_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) = K2 )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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_4063_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: 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,K2)))) = 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),K2)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_4064_take__bit__incr__eq,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) != 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),K2),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),K2)) ) ) ).

% take_bit_incr_eq
tff(fact_4065_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_4066_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K2))) = 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),K2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_4067_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K2: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K2)),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),K2))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_4068_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),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_4069_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K2: 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)),K2)
     => ( 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),K2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),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_4070_take__bit__int__greater__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),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),K2),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),K2)) ) ).

% take_bit_int_greater_eq
tff(fact_4071_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2) = 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),K2),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_4072_and__int__rec,axiom,
    ! [K2: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2)
            & ~ 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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% and_int_rec
tff(fact_4073_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( 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(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_4074_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K2: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> 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),K2),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_4075_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K2: 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,K2))) = 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),K2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_4076_take__bit__minus__small__eq,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K2),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),K2)) = 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)),K2) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_4077_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: 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,K2)))) = 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(K2))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_4078_and__int_Osimps,axiom,
    ! [K2: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = $ite(
        ( member(int,K2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & member(int,L,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2)
            & ~ 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),
              ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2)
              & ~ 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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ).

% and_int.simps
tff(fact_4079_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 = $ite(
            ( member(int,Xb,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,Xaa,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xb)
                & ~ 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),
                  ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xb)
                  & ~ 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),divide_divide(int,Xb,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,Xaa,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.elims
tff(fact_4080_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: 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,K2)))) = 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(K2))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_4081_Diff__insert0,axiom,
    ! [A: $tType,Xb: A,A5: set(A),B5: set(A)] :
      ( ~ member(A,Xb,A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) ) ) ).

% Diff_insert0
tff(fact_4082_insert__Diff1,axiom,
    ! [A: $tType,Xb: A,B5: set(A),A5: set(A)] :
      ( member(A,Xb,B5)
     => ( 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),A5)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) ) ) ).

% insert_Diff1
tff(fact_4083_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A5: 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)),A5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = aa(set(A),set(A),insert(A,A2),A5) ).

% insert_Diff_single
tff(fact_4084_finite__Diff__insert,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,A2),B5)))
    <=> aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) ) ).

% finite_Diff_insert
tff(fact_4085_pred__numeral__inc,axiom,
    ! [K2: num] : pred_numeral(inc(K2)) = aa(num,nat,numeral_numeral(nat),K2) ).

% pred_numeral_inc
tff(fact_4086_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),Xb: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),insert(A,Xb),A5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5)) ) ) ) ) ).

% sum.insert
tff(fact_4087_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),Xb: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),insert(A,Xb),A5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)) ) ) ) ) ).

% prod.insert
tff(fact_4088_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: 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,K2),bot_bot(set(A)))),set_ord_lessThan(A,K2)) = aa(set(A),set(A),insert(A,K2),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_4089_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_4090_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_4091_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_4092_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_4093_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_4094_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_4095_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_4096_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_4097_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_4098_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_4099_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_4100_insert__Diff__if,axiom,
    ! [A: $tType,Xb: A,A5: set(A),B5: 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),A5)),B5) = $ite(member(A,Xb,B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5),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)),A5),B5))) ).

% insert_Diff_if
tff(fact_4101_num__induct,axiom,
    ! [P: fun(num,$o),Xb: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X4: num] :
            ( aa(num,$o,P,X4)
           => aa(num,$o,P,inc(X4)) )
       => aa(num,$o,P,Xb) ) ) ).

% num_induct
tff(fact_4102_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_4103_Diff__insert__absorb,axiom,
    ! [A: $tType,Xb: A,A5: set(A)] :
      ( ~ member(A,Xb,A5)
     => ( 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),A5)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = A5 ) ) ).

% Diff_insert_absorb
tff(fact_4104_Diff__insert2,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,A2),B5)) = 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)),A5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))),B5) ).

% Diff_insert2
tff(fact_4105_insert__Diff,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( member(A,A2,A5)
     => ( 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)),A5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = A5 ) ) ).

% insert_Diff
tff(fact_4106_Diff__insert,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,A2),B5)) = 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)),A5),B5)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_4107_subset__Diff__insert,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),Xb: A,C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),aa(set(A),set(A),insert(A,Xb),C5)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C5))
        & ~ member(A,Xb,A5) ) ) ).

% subset_Diff_insert
tff(fact_4108_inc_Osimps_I1_J,axiom,
    inc(one2) = aa(num,num,bit0,one2) ).

% inc.simps(1)
tff(fact_4109_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : inc(aa(num,num,bit1,Xb)) = aa(num,num,bit0,inc(Xb)) ).

% inc.simps(3)
tff(fact_4110_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : inc(aa(num,num,bit0,Xb)) = aa(num,num,bit1,Xb) ).

% inc.simps(2)
tff(fact_4111_add__One,axiom,
    ! [Xb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2) = inc(Xb) ).

% add_One
tff(fact_4112_inc__BitM__eq,axiom,
    ! [Nb: num] : inc(bitM(Nb)) = aa(num,num,bit0,Nb) ).

% inc_BitM_eq
tff(fact_4113_BitM__inc__eq,axiom,
    ! [Nb: num] : bitM(inc(Nb)) = aa(num,num,bit1,Nb) ).

% BitM_inc_eq
tff(fact_4114_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S3: set(A),P: fun(set(A),$o),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X4: A,S5: set(A)] :
                  ( aa(set(A),$o,finite_finite(A),S5)
                 => ( ! [Y4: A] :
                        ( member(A,Y4,S5)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Y4)),aa(A,B,F3,X4)) )
                   => ( aa(set(A),$o,P,S5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X4),S5)) ) ) )
             => aa(set(A),$o,P,S3) ) ) ) ) ).

% finite_ranking_induct
tff(fact_4115_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B4: A,A7: set(A)] :
                  ( aa(set(A),$o,finite_finite(A),A7)
                 => ( ! [X: A] :
                        ( member(A,X,A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),X) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B4),A7)) ) ) )
             => aa(set(A),$o,P,A5) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4116_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B4: A,A7: set(A)] :
                  ( aa(set(A),$o,finite_finite(A),A7)
                 => ( ! [X: A] :
                        ( member(A,X,A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),B4) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B4),A7)) ) ) )
             => aa(set(A),$o,P,A5) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4117_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_4118_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),insert(A,Xb),A5)) = $ite(member(A,Xb,A5),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5))) ) ) ) ).

% sum.insert_if
tff(fact_4119_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),insert(A,Xb),A5)) = $ite(member(A,Xb,A5),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5))) ) ) ) ).

% prod.insert_if
tff(fact_4120_infinite__remove,axiom,
    ! [A: $tType,S3: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite(A),S3)
     => ~ aa(set(A),$o,finite_finite(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))))) ) ).

% infinite_remove
tff(fact_4121_infinite__coinduct,axiom,
    ! [A: $tType,X6: fun(set(A),$o),A5: set(A)] :
      ( aa(set(A),$o,X6,A5)
     => ( ! [A7: set(A)] :
            ( aa(set(A),$o,X6,A7)
           => ? [X: A] :
                ( member(A,X,A7)
                & ( aa(set(A),$o,X6,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,X),bot_bot(set(A)))))
                  | ~ aa(set(A),$o,finite_finite(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,X),bot_bot(set(A))))) ) ) )
       => ~ aa(set(A),$o,finite_finite(A),A5) ) ) ).

% infinite_coinduct
tff(fact_4122_finite__empty__induct,axiom,
    ! [A: $tType,A5: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(A),$o,P,A5)
       => ( ! [A4: A,A7: set(A)] :
              ( aa(set(A),$o,finite_finite(A),A7)
             => ( member(A,A4,A7)
               => ( aa(set(A),$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_4123_Diff__single__insert,axiom,
    ! [A: $tType,A5: set(A),Xb: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),insert(A,Xb),B5)) ) ).

% Diff_single_insert
tff(fact_4124_subset__insert__iff,axiom,
    ! [A: $tType,A5: set(A),Xb: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),insert(A,Xb),B5))
    <=> $ite(member(A,Xb,A5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)) ) ).

% subset_insert_iff
tff(fact_4125_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_4126_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xb))),aa(set(A),set(A),insert(A,Xb),aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_4127_Compl__insert,axiom,
    ! [A: $tType,Xb: A,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,Xb),A5)) = 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)),A5)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_4128_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_4129_finite__remove__induct,axiom,
    ! [A: $tType,B5: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [A7: set(A)] :
              ( aa(set(A),$o,finite_finite(A),A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B5)
                 => ( ! [X: A] :
                        ( member(A,X,A7)
                       => aa(set(A),$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,X),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B5) ) ) ) ).

% finite_remove_induct
tff(fact_4130_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),B5: set(A)] :
      ( aa(set(A),$o,P,bot_bot(set(A)))
     => ( ( ~ aa(set(A),$o,finite_finite(A),B5)
         => aa(set(A),$o,P,B5) )
       => ( ! [A7: set(A)] :
              ( aa(set(A),$o,finite_finite(A),A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B5)
                 => ( ! [X: A] :
                        ( member(A,X,A7)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A7),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B5) ) ) ) ).

% remove_induct
tff(fact_4131_finite__induct__select,axiom,
    ! [A: $tType,S3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite(A),S3)
     => ( aa(set(A),$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),S3)
             => ( aa(set(A),$o,P,T5)
               => ? [X: A] :
                    ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T5))
                    & aa(set(A),$o,P,aa(set(A),set(A),insert(A,X),T5)) ) ) )
         => aa(set(A),$o,P,S3) ) ) ) ).

% finite_induct_select
tff(fact_4132_psubset__insert__iff,axiom,
    ! [A: $tType,A5: set(A),Xb: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),aa(set(A),set(A),insert(A,Xb),B5))
    <=> $ite(
          member(A,Xb,B5),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5),
          $ite(member(A,Xb,A5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)) ) ) ).

% psubset_insert_iff
tff(fact_4133_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_4134_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_4135_sum__diff1__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A),A2: A] :
      aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(member(A,A2,A5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5)),aa(A,nat,F3,A2)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5)) ).

% sum_diff1_nat
tff(fact_4136_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_4137_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),insert(int,I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_4138_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),Xb: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_4139_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),insert(A,Xb),A5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_4140_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A5: set(A),F3: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(member(A,A2,A5),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),aa(A,B,F3,A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)) ) ) ) ).

% sum_diff1
tff(fact_4141_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),Xb: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_4142_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),insert(A,Xb),A5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xb)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_4143_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B),C3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( 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_jh(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C3)),S3) = $ite(member(A,A2,S3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C3),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)))))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C3),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)))))) ) ) ) ).

% sum.delta_remove
tff(fact_4144_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B),C3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( 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_ji(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C3)),S3) = $ite(member(A,A2,S3),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),C3),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)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C3),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)))))) ) ) ) ).

% prod.delta_remove
tff(fact_4145_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,I,A5)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X4)) )
           => ( aa(set(A),$o,finite_finite(A),A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)) ) ) ) ) ).

% member_le_sum
tff(fact_4146_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A5: set(A),F3: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( aa(A,B,F3,A2) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(member(A,A2,A5),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5),aa(A,B,F3,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ) ) ).

% prod_diff1
tff(fact_4147_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sinh(A,Xb) = zero_zero(A) )
        <=> member(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_4148_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),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% and_nat_unfold
tff(fact_4149_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),
            ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Mb)
            & ~ 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),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% and_nat_rec
tff(fact_4150_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A5: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite(A),A5)
     => ? [H3: fun(A,A)] : bij_betw(A,A,H3,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_4151_and__int_Opsimps,axiom,
    ! [K2: 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),K2),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L) = $ite(
            ( member(int,K2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,L,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2)
                & ~ 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),
                  ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2)
                  & ~ 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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_4152_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 = $ite(
                  ( member(int,Xb,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & member(int,Xaa,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xb)
                      & ~ 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),
                        ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xb)
                        & ~ 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),divide_divide(int,Xb,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),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_4153_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),K2)),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,K2),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_4154_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_4155_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_4156_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_4157_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_4158_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2)),zero_zero(int))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_4159_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_4160_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_4161_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_4162_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_4163_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),zero_zero(nat))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ).

% bit_0
tff(fact_4164_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_4165_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_4166_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) )
            & ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_4167_set__encode__insert,axiom,
    ! [A5: set(nat),Nb: nat] :
      ( aa(set(nat),$o,finite_finite(nat),A5)
     => ( ~ member(nat,Nb,A5)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,Nb),A5)) = 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,A5)) ) ) ) ).

% set_encode_insert
tff(fact_4168_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_4169_bit__and__int__iff,axiom,
    ! [K2: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),L)),Nb)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_4170_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_4171_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_4172_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_4173_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_4174_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_4175_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_4176_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_4177_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_4178_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_4179_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_4180_lessThan__Suc,axiom,
    ! [K2: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),insert(nat,K2),set_ord_lessThan(nat,K2)) ).

% lessThan_Suc
tff(fact_4181_atMost__Suc,axiom,
    ! [K2: nat] : set_ord_atMost(nat,aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K2)),set_ord_atMost(nat,K2)) ).

% atMost_Suc
tff(fact_4182_bit__not__int__iff_H,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K2)),one_one(int))),Nb)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb) ) ).

% bit_not_int_iff'
tff(fact_4183_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_4184_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_4185_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_4186_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_4187_lessThan__nat__numeral,axiom,
    ! [K2: num] : set_ord_lessThan(nat,aa(num,nat,numeral_numeral(nat),K2)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K2)),set_ord_lessThan(nat,pred_numeral(K2))) ).

% lessThan_nat_numeral
tff(fact_4188_atMost__nat__numeral,axiom,
    ! [K2: num] : set_ord_atMost(nat,aa(num,nat,numeral_numeral(nat),K2)) = aa(set(nat),set(nat),insert(nat,aa(num,nat,numeral_numeral(nat),K2)),set_ord_atMost(nat,pred_numeral(K2))) ).

% atMost_nat_numeral
tff(fact_4189_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_4190_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),K2)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_4191_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K2: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_concat_bit(Mb,K2),L)),Nb)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),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_4192_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K2) = aa(int,int,bit_concat_bit(Nb,K2),aa(int,int,uminus_uminus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_4193_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_4194_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,divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),Nb) ) ) ).

% bit_Suc
tff(fact_4195_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
          <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_4196_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)
            <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) )
         => ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_4197_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),M2)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),N) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),N) ) ) ) ).

% int_bit_bound
tff(fact_4198_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)),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_4199_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),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_4200_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_4201_bit__int__def,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb)
    <=> ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),divide_divide(int,K2,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_4202_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ( aa(nat,$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_4203_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
         => ( aa(nat,$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_4204_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ 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_4205_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_1)
     => ( ! [X4: A,N: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N),X4)
           => ? [Y4: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N)),Y4)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N),X4),Y4) ) )
       => ? [F2: fun(nat,A)] :
          ! [N8: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N8),aa(nat,A,F2,N8))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N8),aa(nat,A,F2,N8)),aa(nat,A,F2,aa(nat,nat,suc,N8))) ) ) ) ).

% dependent_nat_choice
tff(fact_4206_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)
          <=> $ite(Nb = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2),aa(nat,$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_4207_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> $ite(Nb = zero_zero(nat),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2),aa(nat,$o,bit_se5641148757651400278ts_bit(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_4208_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K: int,L3: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L3))
           => ( ( ~ ( member(int,K,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L3,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L3,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K),L3) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_4209_set__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K2),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,K2),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_4210_unset__bit__eq,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),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,K2),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_4211_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),K2) = 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,K2),Nb)))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)) ).

% take_bit_Suc_from_most
tff(fact_4212_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_4213_or__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L) = $ite(
        ( ( K2 = 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(
          K2 = zero_zero(int),
          L,
          $ite(L = zero_zero(int),K2,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,K2,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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ).

% or_int_unfold
tff(fact_4214_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( 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_4215_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_jj(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_4216_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_4217_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_4218_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_4219_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_4220_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_4221_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_4222_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_4223_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_4224_or__nonnegative__int__iff,axiom,
    ! [K2: 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),K2),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_4225_or__negative__int__iff,axiom,
    ! [K2: 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),K2),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_4226_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_4227_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_4228_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( ( member(A,Xb,ring_1_Ints(A))
            | 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_4229_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))
        <=> ~ member(A,Xb,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_4230_quotient__of__number_I3_J,axiom,
    ! [K2: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K2)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_4231_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_4232_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_4233_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_4234_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_4235_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_4236_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_4237_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_4238_quotient__of__number_I5_J,axiom,
    ! [K2: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K2))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_4239_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).

% quotient_of_number(4)
tff(fact_4240_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_4241_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_4242_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_4243_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_4244_bit__or__int__iff,axiom,
    ! [K2: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L)),Nb)
    <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb)
        | aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_4245_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_4246_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_add
tff(fact_4247_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_4248_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_4249_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_4250_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C3: 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),C3)) = 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),C3)) ) ).

% or.left_commute
tff(fact_4251_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_4252_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = 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),C3)) ) ).

% or.assoc
tff(fact_4253_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] :
          ( member(A,A2,ring_1_Ints(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_4254_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_4255_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: num] : member(A,aa(num,A,numeral_numeral(A),Nb),ring_1_Ints(A)) ) ).

% Ints_numeral
tff(fact_4256_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K2: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(int,A,ring_1_of_int(A),K2)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_or_eq
tff(fact_4257_divide__rat__def,axiom,
    ! [Q3: rat,R2: rat] : divide_divide(rat,Q3,R2) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Q3),aa(rat,rat,inverse_inverse(rat),R2)) ).

% divide_rat_def
tff(fact_4258_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_diff
tff(fact_4259_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_4260_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_4261_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_4262_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_4263_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_4264_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_4265_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_4266_or__greater__eq,axiom,
    ! [L: int,K2: 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),K2),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L)) ) ).

% or_greater_eq
tff(fact_4267_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_4268_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_4269_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( 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_4270_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_jk(A,fun(A,fun(A,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_4271_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_4272_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( 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_4273_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( dvd_dvd(int,B2,A2)
         => member(A,divide_divide(A,aa(int,A,ring_1_of_int(A),A2),aa(int,A,ring_1_of_int(A),B2)),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_4274_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_jl(A,fun(A,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_4275_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( 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))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
            & dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_or_iff
tff(fact_4276_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_4277_bit__nat__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(int,nat,nat2,K2)),Nb)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb) ) ) ).

% bit_nat_iff
tff(fact_4278_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( aa(A,$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_4279_rat__less__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P2),Q3)
    <=> 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_jn(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_4280_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( ( Xb != zero_zero(A) )
           => aa(A,$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_4281_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( aa(A,$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_4282_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( member(A,Y,ring_1_Ints(A))
           => ( ( Xb = Y )
            <=> aa(A,$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_4283_rat__less__eq__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P2),Q3)
    <=> 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_jp(rat,fun(int,fun(int,$o)),Q3)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_4284_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) )
    <=> member(real,Xb,ring_1_Ints(real)) ) ).

% sin_times_pi_eq_0
tff(fact_4285_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),Xb)) = $ite(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_4286_bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Mb),Nb)
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),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_4287_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_4288_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,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,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)))),aa(int,B,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_4289_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: A] :
          ( ( archimedean_frac(A,Xb) = A2 )
        <=> ( member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xb),A2),ring_1_Ints(A))
            & aa(A,$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_4290_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,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))),aa(int,B,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_4291_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),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))) ) ).

% one_or_eq
tff(fact_4292_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),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))) ) ).

% or_one_eq
tff(fact_4293_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_4294_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_4295_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( 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_4296_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( 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_4297_or__int__rec,axiom,
    ! [K2: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2)
            | ~ 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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% or_int_rec
tff(fact_4298_quotient__of__int,axiom,
    ! [A2: int] : quotient_of(of_int(A2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),one_one(int)) ).

% quotient_of_int
tff(fact_4299_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P2),Q3)) = 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_jr(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_4300_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P2),Q3)) = 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_jt(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_4301_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_4302_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_4303_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_4304_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_4305_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_4306_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_4307_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_4308_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_4309_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_4310_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_4311_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_4312_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_4313_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_4314_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_4315_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_4316_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_4317_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_4318_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_4319_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_4320_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_4321_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_4322_normalize__crossproduct,axiom,
    ! [Q3: int,S2: int,P2: int,R2: int] :
      ( ( Q3 != zero_zero(int) )
     => ( ( S2 != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R2),S2)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S2) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q3) ) ) ) ) ).

% normalize_crossproduct
tff(fact_4323_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_4324_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),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb))) ).

% or_Suc_0_eq
tff(fact_4325_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),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb))) ).

% Suc_0_or_eq
tff(fact_4326_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),
            ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Mb)
            | ~ 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),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% or_nat_rec
tff(fact_4327_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(divide_divide(rat,P2,Q3)) = 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_jv(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_4328_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q3)) = 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_jx(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P2)) ).

% rat_times_code
tff(fact_4329_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),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% or_nat_unfold
tff(fact_4330_Frct__code__post_I5_J,axiom,
    ! [K2: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K2))) = divide_divide(rat,one_one(rat),aa(num,rat,numeral_numeral(rat),K2)) ).

% Frct_code_post(5)
tff(fact_4331_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($o,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb))) ).

% xor_Suc_0_eq
tff(fact_4332_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($o,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb))) ).

% Suc_0_xor_eq
tff(fact_4333_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_4334_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_4335_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_4336_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_4337_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_4338_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_4339_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_4340_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_4341_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_4342_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_4343_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_4344_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_4345_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_4346_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_4347_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_4348_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_4349_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_4350_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_4351_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_4352_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_4353_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C3: 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),C3)) = 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),C3)) ) ).

% xor.left_commute
tff(fact_4354_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_4355_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C3: 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)),C3) = 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),C3)) ) ).

% xor.assoc
tff(fact_4356_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K2: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(int,A,ring_1_of_int(A),K2)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_xor_eq
tff(fact_4357_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_4358_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_4359_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_4360_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( 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))
        <=> ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
          <=> dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2) ) ) ) ).

% even_xor_iff
tff(fact_4361_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_4362_Frct__code__post_I4_J,axiom,
    ! [K2: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K2)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K2) ).

% Frct_code_post(4)
tff(fact_4363_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),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% xor_nat_unfold
tff(fact_4364_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),
            ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Mb) != ~ 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),divide_divide(nat,Mb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,Nb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% xor_nat_rec
tff(fact_4365_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),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))) ) ).

% one_xor_eq
tff(fact_4366_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),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))) ) ).

% xor_one_eq
tff(fact_4367_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_4368_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_4369_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,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_4370_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I6: set(A),F3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(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_jy(set(A),fun(fun(A,B),fun(A,$o)),I6),F3))),F4)
           => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = $ite(member(A,I,I6),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I6)),aa(A,B,F3,I)),groups1027152243600224163dd_sum(A,B,F3,I6)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_4371_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ca(nat,nat)),set_or7035219750837199246ssThan(nat,Mb,Nb)) = 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_4372_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2) ) ).

% push_bit_nonnegative_int_iff
tff(fact_4373_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_4374_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_4375_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_4376_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),A2) ) ).

% push_bit_push_bit
tff(fact_4377_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,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),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ).

% push_bit_and
tff(fact_4378_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,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),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ).

% push_bit_or
tff(fact_4379_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,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),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ).

% push_bit_xor
tff(fact_4380_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa(int,int,bit_concat_bit(Nb,zero_zero(int)),L) = aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),L) ).

% concat_bit_of_zero_1
tff(fact_4381_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_4382_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_4383_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,Mb: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,Mb,Nb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),I)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),I)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Nb) ) ) ) ) ).

% ivl_subset
tff(fact_4384_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_4385_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_4386_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite(A),set_or7035219750837199246ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_4387_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,Nb: A,Mb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),Nb)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I,Mb)),set_or7035219750837199246ssThan(A,I,Nb)) = set_or7035219750837199246ssThan(A,Nb,Mb) ) ) ) ).

% ivl_diff
tff(fact_4388_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)),set_ord_lessThan(A,Nb)),set_ord_lessThan(A,Mb)) = set_or7035219750837199246ssThan(A,Mb,Nb) ) ).

% lessThan_minus_lessThan
tff(fact_4389_xor__nonnegative__int__iff,axiom,
    ! [K2: 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),K2),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_4390_xor__negative__int__iff,axiom,
    ! [K2: 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),K2),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_4391_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,K2: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),K2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K2))) ) ).

% push_bit_Suc_numeral
tff(fact_4392_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_4393_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K2: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2))) = aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K2)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_4394_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: num,K2: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),K2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K2))) ) ).

% push_bit_numeral
tff(fact_4395_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),P2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_at(set(A),fun(fun(A,B),fun(A,$o)),I6),P2)))
         => ( groups1027152243600224163dd_sum(A,B,P2,aa(set(A),set(A),insert(A,I),I6)) = $ite(member(A,I,I6),groups1027152243600224163dd_sum(A,B,P2,I6),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P2,I)),groups1027152243600224163dd_sum(A,B,P2,I6))) ) ) ) ).

% sum.insert'
tff(fact_4396_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,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_4397_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,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)))) ) ).

% push_bit_Suc
tff(fact_4398_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,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_4399_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2))
        <=> ( ( Nb != zero_zero(nat) )
            | dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_4400_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G3,Nb))) ) ).

% sum.op_ivl_Suc
tff(fact_4401_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G3,Nb))) ) ).

% prod.op_ivl_Suc
tff(fact_4402_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [L: num,K2: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K2))) = aa(A,A,bit_se4730199178511100633sh_bit(A,pred_numeral(L)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K2)))) ) ).

% push_bit_minus_numeral
tff(fact_4403_flip__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : bit_se8732182000553998342ip_bit(int,Nb,K2) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),one_one(int))) ).

% flip_bit_int_def
tff(fact_4404_bit__xor__int__iff,axiom,
    ! [K2: int,L: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L)),Nb)
    <=> ~ ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb)
        <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_4405_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
            <=> ( ( A2 = C3 )
                & ( B2 = D3 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4406_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
             => ( A2 = C3 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4407_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C3,D3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D3)
             => ( B2 = D3 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4408_push__bit__nat__eq,axiom,
    ! [Nb: nat,K2: int] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Nb),aa(int,nat,nat2,K2)) = aa(int,nat,nat2,aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K2)) ).

% push_bit_nat_eq
tff(fact_4409_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K2: int] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(int,A,ring_1_of_int(A),K2)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K2)) ) ).

% push_bit_of_int
tff(fact_4410_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,Mb: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(nat,A,semiring_1_of_nat(A),Mb)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Nb),Mb)) ) ).

% push_bit_of_nat
tff(fact_4411_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Mb),Nb)) = aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_push_bit
tff(fact_4412_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,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),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B2)) ) ).

% push_bit_add
tff(fact_4413_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)) ) ).

% push_bit_minus
tff(fact_4414_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4415_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite(A),set_or7035219750837199246ssThan(A,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_4416_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) )
    <=> ! [X3: nat] :
          ( member(nat,X3,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X3) ) ) ).

% all_nat_less_eq
tff(fact_4417_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) )
    <=> ? [X3: nat] :
          ( member(nat,X3,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X3) ) ) ).

% ex_nat_less_eq
tff(fact_4418_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_4419_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_4420_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,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)),aa(A,A,bit_se4730199178511100633sh_bit(A,Mb),A2)) ) ).

% push_bit_take_bit
tff(fact_4421_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)) = aa(A,A,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_4422_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),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Mb),one_one(nat))) ).

% set_bit_nat_def
tff(fact_4423_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),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Mb),one_one(nat))) ).

% flip_bit_nat_def
tff(fact_4424_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_4425_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_4426_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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_fl(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_4427_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fu(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_4428_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_jz(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% take_bit_sum
tff(fact_4429_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I6)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ka(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H),I6) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G3,I6)),groups1027152243600224163dd_sum(A,B,H,I6)) ) ) ) ).

% sum.distrib_triv'
tff(fact_4430_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C3: A,B2: A,D3: A,G3: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C3 )
         => ( ( B2 = D3 )
           => ( ! [X4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),D3)
                   => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),set_or7035219750837199246ssThan(A,C3,D3)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_4431_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C3: A,B2: A,D3: A,G3: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C3 )
         => ( ( B2 = D3 )
           => ( ! [X4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),D3)
                   => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),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,C3,D3)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_4432_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,P2: nat,G3: fun(nat,A)] :
          ( aa(nat,$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,G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,Nb,P2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,Mb,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_4433_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,P2: nat,F3: fun(nat,A)] :
          ( aa(nat,$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,F3),set_or7035219750837199246ssThan(nat,Mb,P2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,Mb,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,Nb,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_4434_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,P2: nat,G3: fun(nat,A)] :
          ( aa(nat,$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),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Nb,P2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_4435_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_4436_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K2: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,Mb),K2)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_4437_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_4438_bit__push__bit__iff__nat,axiom,
    ! [Mb: nat,Q3: nat,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Mb),Q3)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Q3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_4439_concat__bit__eq,axiom,
    ! [Nb: nat,K2: int,L: int] : aa(int,int,bit_concat_bit(Nb,K2),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),L)) ).

% concat_bit_eq
tff(fact_4440_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),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_4441_concat__bit__def,axiom,
    ! [Nb: nat,K2: int,L: int] : aa(int,int,bit_concat_bit(Nb,K2),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K2)),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),L)) ).

% concat_bit_def
tff(fact_4442_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),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_4443_set__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Nb),K2) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),one_one(int))) ).

% set_bit_int_def
tff(fact_4444_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T2: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
               => ( aa(A,B,G3,X4) = zero_zero(B) ) )
           => ( ! [X4: A] :
                  ( member(A,X4,S3)
                 => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
             => ( groups1027152243600224163dd_sum(A,B,G3,T2) = groups1027152243600224163dd_sum(A,B,H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_4445_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T2: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
               => ( aa(A,B,H,I2) = zero_zero(B) ) )
           => ( ! [X4: A] :
                  ( member(A,X4,S3)
                 => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
             => ( groups1027152243600224163dd_sum(A,B,G3,S3) = groups1027152243600224163dd_sum(A,B,H,T2) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_4446_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
               => ( aa(A,B,G3,X4) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G3,T2) = groups1027152243600224163dd_sum(A,B,G3,S3) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_4447_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
               => ( aa(A,B,G3,X4) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G3,S3) = groups1027152243600224163dd_sum(A,B,G3,T2) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_4448_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4449_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4450_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K2: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_4451_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G3,Nb)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_4452_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_4453_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_4454_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I6: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_at(set(A),fun(fun(A,B),fun(A,$o)),I6),G3)))
         => ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_at(set(A),fun(fun(A,B),fun(A,$o)),I6),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ka(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H),I6) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G3,I6)),groups1027152243600224163dd_sum(A,B,H,I6)) ) ) ) ) ).

% sum.distrib'
tff(fact_4455_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_4456_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G3,Nb)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4457_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$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),G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),Nb))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4458_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G3: fun(nat,A)] :
          ( aa(nat,$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),G3),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),G3),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_4459_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Nb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% sum.last_plus
tff(fact_4460_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$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),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Nb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))) ) ) ) ).

% prod.last_plus
tff(fact_4461_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,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),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),A2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% push_bit_double
tff(fact_4462_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Mb: nat,Nb: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bv(fun(nat,A),fun(nat,A),F3)),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,Nb)),aa(nat,A,F3,Mb)) ) ) ) ).

% sum_Suc_diff'
tff(fact_4463_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),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_4464_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_4465_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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_kb(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% sum.atLeastLessThan_rev
tff(fact_4466_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_kc(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_ge(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% sum.nested_swap
tff(fact_4467_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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_kd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or7035219750837199246ssThan(nat,Nb,Mb)) ) ).

% prod.atLeastLessThan_rev
tff(fact_4468_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_ke(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_fo(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% prod.nested_swap
tff(fact_4469_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),K2: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kf(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2))) ) ).

% sum.nat_group
tff(fact_4470_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),K2: 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_kg(fun(nat,A),fun(nat,fun(nat,A)),G3),K2)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K2))) ) ).

% prod.nat_group
tff(fact_4471_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_4472_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_4473_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G3,Nb))) ) ).

% sum.head_if
tff(fact_4474_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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),G3),set_or7035219750837199246ssThan(nat,Mb,Nb))),aa(nat,A,G3,Nb))) ) ).

% prod.head_if
tff(fact_4475_push__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K2) = aa(int,int,aa(int,fun(int,int),times_times(int),K2),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_4476_push__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,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_4477_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,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_4478_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( 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)
         => ~ ! [B4: A] : A2 != aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),B4) ) ) ).

% exp_dvdE
tff(fact_4479_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_4480_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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_bs(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4481_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat,Mb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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_fy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Nb),Mb)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),Mb)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4482_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_gm(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_prod
tff(fact_4483_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K2: num] :
      set_or7035219750837199246ssThan(nat,Mb,aa(num,nat,numeral_numeral(nat),K2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),pred_numeral(K2)),aa(set(nat),set(nat),insert(nat,pred_numeral(K2)),set_or7035219750837199246ssThan(nat,Mb,pred_numeral(K2))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_4484_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_4485_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
        <=> ! [E4: real] :
              ( aa(real,$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,F3),set_or7035219750837199246ssThan(nat,M5,N4)))),E4) ) ) ) ) ).

% summable_Cauchy
tff(fact_4486_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa(int,int,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_4487_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A),S2: A,K2: nat] :
          ( sums(A,F3,S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_kh(fun(nat,A),fun(nat,fun(nat,A)),F3),K2),S2) ) ) ) ).

% sums_group
tff(fact_4488_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_4489_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)),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_4490_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),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),K2),Nb)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),K2))) ) ) ) ).

% fact_split
tff(fact_4491_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ki(nat,fun(nat,fun(nat,A)),K2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_4492_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),K2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_kj(A,fun(nat,fun(nat,A)),A2),K2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_altdef_of_nat
tff(fact_4493_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K2: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K2)),aa(nat,A,gbinomial(A,A2),K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_kk(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_mult_fact
tff(fact_4494_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K2: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K2)),semiring_char_0_fact(A,K2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_kk(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)) ) ).

% gbinomial_mult_fact'
tff(fact_4495_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K2: nat] : aa(nat,A,gbinomial(A,A2),K2) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ha(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K2)),semiring_char_0_fact(A,K2)) ) ).

% gbinomial_prod_rev
tff(fact_4496_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I6: set(A),F3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_jy(set(A),fun(fun(A,B),fun(A,$o)),I6),F3)))
         => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = $ite(member(A,I,I6),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I6)),aa(A,B,F3,I)),groups1027152243600224163dd_sum(A,B,F3,I6)) ) ) ) ).

% sum_diff1'
tff(fact_4497_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),aa(A,A,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_4498_xor__int__rec,axiom,
    ! [K2: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2) != ~ 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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% xor_int_rec
tff(fact_4499_sum__power2,axiom,
    ! [K2: 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),K2)) = 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))),K2)),one_one(nat)) ).

% sum_power2
tff(fact_4500_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,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_kl(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_4501_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,I2)),aa(nat,A,A2,J2)) ) )
         => ( ! [I2: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Nb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I2)) ) )
           => aa(A,$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_km(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_4502_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),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,I2)),aa(nat,nat,A2,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),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,I2)) ) )
       => 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_kn(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_4503_xor__int__unfold,axiom,
    ! [K2: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),L) = $ite(
        K2 = 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),K2),
          $ite(
            K2 = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K2,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,K2,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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_4504_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_4505_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_4506_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_4507_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_4508_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_4509_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_4510_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_4511_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_4512_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_4513_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_4514_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_4515_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_4516_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_4517_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_4518_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_4519_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_4520_not__negative__int__iff,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2) ) ).

% not_negative_int_iff
tff(fact_4521_not__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_4522_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_4523_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_ri4277139882892585799ns_not(A),A2))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ).

% even_not_iff
tff(fact_4524_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,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_4525_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_4526_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_4527_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_4528_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K2: num] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),K2))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),K2)) ) ).

% of_int_not_numeral
tff(fact_4529_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),K2)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(int,A,ring_1_of_int(A),K2)) ) ).

% of_int_not_eq
tff(fact_4530_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_4531_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_4532_bit__not__int__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K2)),Nb)
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K2),Nb) ) ).

% bit_not_int_iff
tff(fact_4533_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_4534_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_4535_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_4536_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_4537_or__int__def,axiom,
    ! [K2: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K2),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),K2)),aa(int,int,bit_ri4277139882892585799ns_not(int),L))) ).

% or_int_def
tff(fact_4538_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_4539_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_4540_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_4541_not__int__def,axiom,
    ! [K2: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K2)),one_one(int)) ).

% not_int_def
tff(fact_4542_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_4543_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_4544_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_4545_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_4546_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_4547_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_4548_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_4549_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_4550_unset__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Nb),K2) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K2),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),one_one(int)))) ).

% unset_bit_int_def
tff(fact_4551_xor__int__def,axiom,
    ! [K2: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),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),K2),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),K2)),L)) ).

% xor_int_def
tff(fact_4552_not__int__div__2,axiom,
    ! [K2: int] : divide_divide(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = aa(int,int,bit_ri4277139882892585799ns_not(int),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) ).

% not_int_div_2
tff(fact_4553_even__not__iff__int,axiom,
    ! [K2: int] :
      ( dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,bit_ri4277139882892585799ns_not(int),K2))
    <=> ~ dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2) ) ).

% even_not_iff_int
tff(fact_4554_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_4555_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_4556_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_4557_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_4558_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_4559_bit__minus__int__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K2)),Nb)
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K2),one_one(int)))),Nb) ) ).

% bit_minus_int_iff
tff(fact_4560_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_4561_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_4562_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_4563_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_4564_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_4565_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Mb: nat,Nb: nat] : aa(A,A,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_4566_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),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_4567_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_4568_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_4569_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_4570_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_4571_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_4572_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_4573_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_4574_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_4575_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_4576_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_4577_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_4578_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_4579_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_4580_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_4581_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_4582_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_4583_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_4584_not__int__rec,axiom,
    ! [K2: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K2))),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),divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% not_int_rec
tff(fact_4585_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_4586_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),aa(list(vEBT_VEBT),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_4587_Cauchy__iff2,axiom,
    ! [X6: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X6)
    <=> ! [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,X6,M5)),aa(nat,real,X6,N4)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_4588_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D3: nat] :
      ( vEBT_VEBT_valid(Ta,D3)
    <=> vEBT_invar_vebt(Ta,D3) ) ).

% valid_eq
tff(fact_4589_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D3: nat] :
      ( vEBT_VEBT_valid(Ta,D3)
     => vEBT_invar_vebt(Ta,D3) ) ).

% valid_eq2
tff(fact_4590_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D3: nat] :
      ( vEBT_invar_vebt(Ta,D3)
     => vEBT_VEBT_valid(Ta,D3) ) ).

% valid_eq1
tff(fact_4591_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D3: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu2),(Uv2)),D3)
    <=> ( D3 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_4592_size__list__estimation,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(A,nat,F3,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(list(A),nat,size_list(A,F3),Xs)) ) ) ).

% size_list_estimation
tff(fact_4593_size__list__estimation_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F3,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(list(A),nat,size_list(A,F3),Xs)) ) ) ).

% size_list_estimation'
tff(fact_4594_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat),G3: fun(A,nat)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,X4)),aa(A,nat,G3,X4)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_list(A,F3),Xs)),aa(list(A),nat,size_list(A,G3),Xs)) ) ).

% size_list_pointwise
tff(fact_4595_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E4: real] :
              ( aa(real,$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,X6,M5)),aa(nat,A,X6,N4)))),E4) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_4596_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$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,X6,M)),aa(nat,A,X6,N)))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI
tff(fact_4597_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => ? [M7: nat] :
              ! [M2: nat] :
                ( aa(nat,$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,X6,M2)),aa(nat,A,X6,N8)))),E3) ) ) ) ) ) ).

% CauchyD
tff(fact_4598_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),aa(list(vEBT_VEBT),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_4599_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_4600_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_4601_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_4602_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_4603_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_4604_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% ord_eq_le_trans
tff(fact_4605_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( B2 = C3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% ord_le_eq_trans
tff(fact_4606_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_4607_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% order.trans
tff(fact_4608_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_4609_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B4) )
         => ( ! [A4: A,B4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B4),A4)
               => aa(A,$o,aa(A,fun(A,$o),P,A4),B4) )
           => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ).

% linorder_wlog
tff(fact_4610_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_4611_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_4612_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% dual_order.trans
tff(fact_4613_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_4614_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_4615_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C3)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X4)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_subst1
tff(fact_4616_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,B2)),C3)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_subst2
tff(fact_4617_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_4618_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_4619_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A2 = aa(B,A,F3,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C3)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X4)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_4620_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_4621_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_4622_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_4623_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xb: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xb) ) ).

% lt_ex
tff(fact_4624_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_4625_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
         => ? [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),Y) ) ) ) ).

% dense
tff(fact_4626_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_4627_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_4628_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% ord_eq_less_trans
tff(fact_4629_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( B2 = C3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% ord_less_eq_trans
tff(fact_4630_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [X4: A] :
              ( ! [Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X4)
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X4) )
         => aa(A,$o,P,A2) ) ) ).

% less_induct
tff(fact_4631_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_4632_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_4633_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_4634_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_4635_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
        <=> ? [N4: A] :
              ( aa(A,$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_4636_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B4) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P,A4),A4)
           => ( ! [A4: A,B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),P,A4),B4) )
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_4637_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% order.strict_trans
tff(fact_4638_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_4639_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% dual_order.strict_trans
tff(fact_4640_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_4641_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_4642_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_4643_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_4644_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_4645_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_4646_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_4647_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A2 = aa(B,A,F3,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C3)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X4)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_4648_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_4649_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_4650_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C3)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X4)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_less_subst1
tff(fact_4651_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,B2)),C3)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_less_subst2
tff(fact_4652_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_4653_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_4654_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_4655_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_4656_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_4657_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_4658_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_4659_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_4660_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_4661_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_4662_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_4663_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_4664_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X4) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_ge
tff(fact_4665_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_le
tff(fact_4666_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_4667_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_4668_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_4669_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_4670_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% order.strict_trans1
tff(fact_4671_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% order.strict_trans2
tff(fact_4672_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_4673_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_4674_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_4675_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_4676_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_4677_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_4678_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_4679_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_4680_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_4681_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_4682_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_4683_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_4684_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_4685_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_4686_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_4687_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_4688_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_4689_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_4690_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_4691_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C3)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X4)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_4692_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,B2)),C3)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_le_less_subst2
tff(fact_4693_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C3)
           => ( ! [X4: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X4)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_4694_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,B2)),C3)
           => ( ! [X4: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_less_le_subst2
tff(fact_4695_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_4696_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_4697_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_4698_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_4699_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_4700_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_4701_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_4702_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_4703_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_4704_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_4705_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,G3,Xb)) ) ) ).

% le_funD
tff(fact_4706_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,G3,Xb)) ) ) ).

% le_funE
tff(fact_4707_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3) ) ) ).

% le_funI
tff(fact_4708_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
        <=> ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) ) ) ).

% le_fun_def
tff(fact_4709_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_4710_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_ko(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_4711_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : re(csqrt(Z)) = aa(real,real,sqrt,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_4712_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_4713_minus__integer__code_I1_J,axiom,
    ! [K2: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),K2),zero_zero(code_integer)) = K2 ).

% minus_integer_code(1)
tff(fact_4714_subseqs__refl,axiom,
    ! [A: $tType,Xs: list(A)] : member(list(A),Xs,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) ).

% subseqs_refl
tff(fact_4715_sgn__integer__code,axiom,
    ! [K2: code_integer] :
      aa(code_integer,code_integer,sgn_sgn(code_integer),K2) = $ite(
        K2 = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K2),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),one_one(code_integer)) ) ).

% sgn_integer_code
tff(fact_4716_times__integer__code_I1_J,axiom,
    ! [K2: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),K2),zero_zero(code_integer)) = zero_zero(code_integer) ).

% times_integer_code(1)
tff(fact_4717_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_4718_one__complex_Osimps_I1_J,axiom,
    re(one_one(complex)) = one_one(real) ).

% one_complex.simps(1)
tff(fact_4719_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,Xb: complex] : re(real_V8093663219630862766scaleR(complex,R2,Xb)) = aa(real,real,aa(real,fun(real,real),times_times(real),R2),re(Xb)) ).

% scaleR_complex.simps(1)
tff(fact_4720_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_4721_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_4722_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_4723_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_4724_csqrt_Ocode,axiom,
    ! [Z: complex] :
      csqrt(Z) = complex2(aa(real,real,sqrt,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,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_4725_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,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_4726_integer__of__int__code,axiom,
    ! [K2: int] :
      code_integer_of_int(K2) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K2))),
        $ite(
          K2 = 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(divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),
            $ite(modulo_modulo(int,K2,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_4727_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_4728_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_4729_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_4730_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),real_Vector_of_real(complex,aa(real,real,sqrt,aa(real,real,abs_abs(real),re(Xb))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_4731_imaginary__unit_Osimps_I2_J,axiom,
    im(imaginary_unit) = one_one(real) ).

% imaginary_unit.simps(2)
tff(fact_4732_one__complex_Osimps_I2_J,axiom,
    im(one_one(complex)) = zero_zero(real) ).

% one_complex.simps(2)
tff(fact_4733_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_4734_one__integer__def,axiom,
    one_one(code_integer) = code_integer_of_int(one_one(int)) ).

% one_integer_def
tff(fact_4735_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_4736_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,Xb: complex] : im(real_V8093663219630862766scaleR(complex,R2,Xb)) = aa(real,real,aa(real,fun(real,real),times_times(real),R2),im(Xb)) ).

% scaleR_complex.simps(2)
tff(fact_4737_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_4738_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_4739_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_4740_scaleR__complex_Ocode,axiom,
    ! [R2: real,Xb: complex] : real_V8093663219630862766scaleR(complex,R2,Xb) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R2),re(Xb)),aa(real,real,aa(real,fun(real,real),times_times(real),R2),im(Xb))) ).

% scaleR_complex.code
tff(fact_4741_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_4742_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_4743_Re__exp,axiom,
    ! [Z: complex] : re(exp(complex,Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),exp(real,re(Z))),cos(real,im(Z))) ).

% Re_exp
tff(fact_4744_Im__exp,axiom,
    ! [Z: complex] : im(exp(complex,Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),exp(real,re(Z))),sin(real,im(Z))) ).

% Im_exp
tff(fact_4745_fun__complex__eq,axiom,
    ! [A: $tType,F3: fun(A,complex),X: A] : aa(A,complex,F3,X) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,re(aa(A,complex,F3,X)))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,im(aa(A,complex,F3,X))))) ).

% fun_complex_eq
tff(fact_4746_complex__eq,axiom,
    ! [A2: complex] : A2 = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,re(A2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,im(A2)))) ).

% complex_eq
tff(fact_4747_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_4748_exp__eq__polar,axiom,
    ! [Z: complex] : exp(complex,Z) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,exp(real,re(Z)))),cis(im(Z))) ).

% exp_eq_polar
tff(fact_4749_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_4750_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_4751_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_4752_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_4753_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_4754_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : re(aa(complex,complex,inverse_inverse(complex),Xb)) = 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_4755_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_4756_Re__divide,axiom,
    ! [Xb: complex,Y: complex] : re(divide_divide(complex,Xb,Y)) = 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_4757_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_4758_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_4759_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : im(aa(complex,complex,inverse_inverse(complex),Xb)) = 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_4760_Im__divide,axiom,
    ! [Xb: complex,Y: complex] : im(divide_divide(complex,Xb,Y)) = 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_4761_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_4762_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),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),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_4763_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : aa(complex,complex,inverse_inverse(complex),Xb) = complex2(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))))),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_4764_Complex__divide,axiom,
    ! [Xb: complex,Y: complex] : divide_divide(complex,Xb,Y) = complex2(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))))),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_4765_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ! [X4: list(A)] :
          ( member(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X4) = 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_4766_Im__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member(complex,R2,real_Vector_Reals(complex))
     => ( im(divide_divide(complex,R2,Z)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),re(R2))),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_4767_Re__Reals__divide,axiom,
    ! [R2: complex,Z: complex] :
      ( member(complex,R2,real_Vector_Reals(complex))
     => ( re(divide_divide(complex,R2,Z)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),re(R2)),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_4768_real__eq__imaginary__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( member(complex,Y,real_Vector_Reals(complex))
     => ( 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_4769_imaginary__eq__real__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( member(complex,Y,real_Vector_Reals(complex))
     => ( 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_4770_Reals__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,Nb: nat] :
          ( member(A,A2,real_Vector_Reals(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),real_Vector_Reals(A)) ) ) ).

% Reals_power
tff(fact_4771_Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,divide_divide(A,A2,B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_divide
tff(fact_4772_Reals__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : member(A,aa(num,A,numeral_numeral(A),W),real_Vector_Reals(A)) ) ).

% Reals_numeral
tff(fact_4773_Reals__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_mult
tff(fact_4774_Reals__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,one_one(A),real_Vector_Reals(A)) ) ).

% Reals_1
tff(fact_4775_Reals__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_add
tff(fact_4776_Reals__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),real_Vector_Reals(A)) ) ) ) ).

% Reals_diff
tff(fact_4777_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,real_Vector_Reals(A))
         => ( member(A,B2,real_Vector_Reals(A))
           => ( ( B2 != zero_zero(A) )
             => member(A,divide_divide(A,A2,B2),real_Vector_Reals(A)) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_4778_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G3: fun(nat,complex),N3: nat,F3: fun(nat,A)] :
          ( summable(complex,G3)
         => ( ! [N: nat] : member(complex,aa(nat,complex,G3,N),real_Vector_Reals(complex))
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(nat,complex,G3,N)))
             => ( ! [N: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N))),real_V7770717601297561774m_norm(complex,aa(nat,complex,G3,N))) )
               => summable(A,F3) ) ) ) ) ) ).

% series_comparison_complex
tff(fact_4779_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),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_4780_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),$o),aTP_Lamp_kp(nat,fun(list(A),fun(list(A),$o)),Nb),Xs)) ).

% set_n_lists
tff(fact_4781_complex__mult__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) = 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_4782_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_4783_complex__cnj__one,axiom,
    cnj(one_one(complex)) = one_one(complex) ).

% complex_cnj_one
tff(fact_4784_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( cnj(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% complex_cnj_one_iff
tff(fact_4785_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_4786_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_4787_Re__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( re(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_4788_Im__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( im(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_4789_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_4790_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),Nb: nat,Xs: list(A)] :
      ( 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_4791_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(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_4792_Re__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),re(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_4793_Re__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(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_4794_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(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_4795_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(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_4796_Im__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(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_4797_Im__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),im(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_4798_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(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_4799_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_4800_complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(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(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_4801_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_4802_complex__norm__square,axiom,
    ! [Z: 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_4803_complex__add__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Z),cnj(Z)) = 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_4804_complex__div__cnj,axiom,
    ! [A2: complex,B2: complex] : divide_divide(complex,A2,B2) = divide_divide(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)),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_4805_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)) = 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_4806_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_4807_int__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_int_of_integer(K2) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K2),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K2))),
        $ite(K2 = 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_kq(code_integer,fun(code_integer,int))),code_divmod_integer(K2,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ) ).

% int_of_integer_code
tff(fact_4808_num__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_num_of_integer(K2) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K2),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_kr(code_integer,fun(code_integer,num))),code_divmod_integer(K2,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% num_of_integer_code
tff(fact_4809_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_4810_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_4811_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_4812_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_4813_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_4814_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_4815_nat__of__integer__code,axiom,
    ! [K2: code_integer] :
      code_nat_of_integer(K2) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K2),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_ks(code_integer,fun(code_integer,nat))),code_divmod_integer(K2,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% nat_of_integer_code
tff(fact_4816_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( dvd_dvd(B,aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5))
          <=> 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_kt(set(A),fun(fun(A,B),fun(A,$o)),A5),F3)))) ) ) ) ).

% even_sum_iff
tff(fact_4817_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,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      (B4),
                      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),(B4))) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [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_4818_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_ai(nat,fun(nat,$o)),Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_4819_card__atMost,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),set_ord_atMost(nat,U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_4820_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_4821_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_ah(nat,fun(nat,$o)),Nb))) = aa(nat,nat,suc,Nb) ).

% card_Collect_le_nat
tff(fact_4822_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_4823_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ku(A,fun(B,A),Y)),A5) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(set(B),nat,finite_card(B),A5)) ) ).

% prod_constant
tff(fact_4824_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_4825_card__insert__disjoint,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( ~ member(A,Xb,A5)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),A5)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A5)) ) ) ) ).

% card_insert_disjoint
tff(fact_4826_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y: A,A5: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_kv(A,fun(B,A),Y)),A5) = 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),A5))),Y) ) ).

% sum_constant
tff(fact_4827_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: set(A)] :
      ( member(A,A2,A5)
     => ( ~ member(A,A2,B5)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,A2),B5))) = 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)),A5),B5))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_4828_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_4829_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B5: set(A),A5: set(B),R2: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( ! [A4: B] :
            ( member(B,A4,A5)
           => ? [B9: A] :
                ( member(A,B9,B5)
                & aa(A,$o,aa(B,fun(A,$o),R2,A4),B9) ) )
       => ( ! [A12: B,A23: B,B4: A] :
              ( member(B,A12,A5)
             => ( member(B,A23,A5)
               => ( member(A,B4,B5)
                 => ( aa(A,$o,aa(B,fun(A,$o),R2,A12),B4)
                   => ( aa(A,$o,aa(B,fun(A,$o),R2,A23),B4)
                     => ( A12 = A23 ) ) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_4830_card__insert__le,axiom,
    ! [A: $tType,A5: set(A),Xb: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),A5))) ).

% card_insert_le
tff(fact_4831_card__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( 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_af(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A5)),Nb) ) ) ).

% card_lists_length_eq
tff(fact_4832_card__eq__sum,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),nat,finite_card(A),A5) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_kw(A,nat)),A5) ).

% card_eq_sum
tff(fact_4833_card__2__iff_H,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S3) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X3: A] :
          ( member(A,X3,S3)
          & ? [Xa2: A] :
              ( member(A,Xa2,S3)
              & ( X3 != Xa2 )
              & ! [Xb4: A] :
                  ( member(A,Xb4,S3)
                 => ( ( Xb4 = X3 )
                    | ( Xb4 = Xa2 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_4834_card__ge__0__finite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5))
     => aa(set(A),$o,finite_finite(A),A5) ) ).

% card_ge_0_finite
tff(fact_4835_card__insert__if,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),A5)) = $ite(member(A,Xb,A5),aa(set(A),nat,finite_card(A),A5),aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A5))) ) ) ).

% card_insert_if
tff(fact_4836_card__Suc__eq__finite,axiom,
    ! [A: $tType,A5: set(A),K2: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K2) )
    <=> ? [B6: A,B10: set(A)] :
          ( ( A5 = aa(set(A),set(A),insert(A,B6),B10) )
          & ~ member(A,B6,B10)
          & ( aa(set(A),nat,finite_card(A),B10) = K2 )
          & aa(set(A),$o,finite_finite(A),B10) ) ) ).

% card_Suc_eq_finite
tff(fact_4837_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),$o),ord_less_eq(set(A)),G4),F4)
         => ( aa(set(A),$o,finite_finite(A),G4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G4)),C5) ) )
     => ( aa(set(A),$o,finite_finite(A),F4)
        & aa(nat,$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_4838_card__seteq,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B5)),aa(set(A),nat,finite_card(A),A5))
         => ( A5 = B5 ) ) ) ) ).

% card_seteq
tff(fact_4839_card__mono,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ).

% card_mono
tff(fact_4840_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Nb: nat,S3: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),S3))
     => ~ ! [T5: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T5),S3)
           => ( ( aa(set(A),nat,finite_card(A),T5) = Nb )
             => ~ aa(set(A),$o,finite_finite(A),T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4841_card__less__sym__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(A),$o,finite_finite(A),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))
         => aa(nat,$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)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4842_card__le__sym__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(A),$o,finite_finite(A),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))
         => aa(nat,$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)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4843_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_4844_card__1__singletonE,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = one_one(nat) )
     => ~ ! [X4: A] : A5 != aa(set(A),set(A),insert(A,X4),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_4845_psubset__card__mono,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ).

% psubset_card_mono
tff(fact_4846_card__less__Suc2,axiom,
    ! [M6: set(nat),I: nat] :
      ( ~ 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_kx(set(nat),fun(nat,fun(nat,$o)),M6),I))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ky(set(nat),fun(nat,fun(nat,$o)),M6),I))) ) ) ).

% card_less_Suc2
tff(fact_4847_card__less__Suc,axiom,
    ! [M6: set(nat),I: nat] :
      ( 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_kx(set(nat),fun(nat,fun(nat,$o)),M6),I)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ky(set(nat),fun(nat,fun(nat,$o)),M6),I))) ) ) ).

% card_less_Suc
tff(fact_4848_card__less,axiom,
    ! [M6: set(nat),I: nat] :
      ( 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_ky(set(nat),fun(nat,fun(nat,$o)),M6),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_4849_sum__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A)] : aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_kz(fun(A,nat),fun(A,nat),F3)),A5) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5)),aa(set(A),nat,finite_card(A),A5)) ).

% sum_Suc
tff(fact_4850_subset__card__intvl__is__intvl,axiom,
    ! [A5: set(nat),K2: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A5),set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(set(nat),nat,finite_card(nat),A5))))
     => ( A5 = set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),aa(set(nat),nat,finite_card(nat),A5))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_4851_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S3: set(A),T2: set(B),R: fun(A,fun(B,$o)),K2: nat] :
      ( aa(set(A),$o,finite_finite(A),S3)
     => ( aa(set(B),$o,finite_finite(B),T2)
       => ( ! [X4: B] :
              ( member(B,X4,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_la(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S3),R),X4))) = K2 ) )
         => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_lc(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T2),R)),S3) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),aa(set(B),nat,finite_card(B),T2)) ) ) ) ) ).

% sum_multicount
tff(fact_4852_real__of__card,axiom,
    ! [A: $tType,A5: set(A)] : aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),A5)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_ld(A,real)),A5) ).

% real_of_card
tff(fact_4853_nat__of__integer__code__post_I3_J,axiom,
    ! [K2: num] : code_nat_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K2)) = aa(num,nat,numeral_numeral(nat),K2) ).

% nat_of_integer_code_post(3)
tff(fact_4854_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_4855_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A5: set(A),F3: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),K5) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),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),A5))),K5)) ) ) ).

% sum_bounded_above
tff(fact_4856_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A5: set(A),K5: B,F3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K5),aa(A,B,F3,I2)) )
         => aa(B,$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),A5))),K5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)) ) ) ).

% sum_bounded_below
tff(fact_4857_card__gt__0__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5))
    <=> ( ( A5 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite(A),A5) ) ) ).

% card_gt_0_iff
tff(fact_4858_card__Suc__eq,axiom,
    ! [A: $tType,A5: set(A),K2: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K2) )
    <=> ? [B6: A,B10: set(A)] :
          ( ( A5 = aa(set(A),set(A),insert(A,B6),B10) )
          & ~ member(A,B6,B10)
          & ( aa(set(A),nat,finite_card(A),B10) = K2 )
          & ( ( K2 = zero_zero(nat) )
           => ( B10 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_4859_card__eq__SucD,axiom,
    ! [A: $tType,A5: set(A),K2: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K2) )
     => ? [B4: A,B8: set(A)] :
          ( ( A5 = aa(set(A),set(A),insert(A,B4),B8) )
          & ~ member(A,B4,B8)
          & ( aa(set(A),nat,finite_card(A),B8) = K2 )
          & ( ( K2 = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_4860_card__1__singleton__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X3: A] : A5 = aa(set(A),set(A),insert(A,X3),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_4861_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X3: A] :
            ( member(A,X3,A5)
           => ! [Xa2: A] :
                ( member(A,Xa2,A5)
               => ( X3 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4862_card__le__Suc__iff,axiom,
    ! [A: $tType,Nb: nat,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(set(A),nat,finite_card(A),A5))
    <=> ? [A6: A,B10: set(A)] :
          ( ( A5 = aa(set(A),set(A),insert(A,A6),B10) )
          & ~ member(A,A6,B10)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(set(A),nat,finite_card(A),B10))
          & aa(set(A),$o,finite_finite(A),B10) ) ) ).

% card_le_Suc_iff
tff(fact_4863_card__Diff1__le,axiom,
    ! [A: $tType,A5: set(A),Xb: A] : aa(nat,$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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)) ).

% card_Diff1_le
tff(fact_4864_card__Diff__subset,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ).

% card_Diff_subset
tff(fact_4865_card__psubset,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5) ) ) ) ).

% card_psubset
tff(fact_4866_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => 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),A5)),aa(set(A),nat,finite_card(A),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))) ) ).

% diff_card_le_card_Diff
tff(fact_4867_card__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( 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_ag(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A5))),set_ord_atMost(nat,Nb)) ) ) ).

% card_lists_length_le
tff(fact_4868_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M6: set(A)] :
      ( aa(set(A),$o,finite_finite(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_4869_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_al(nat,fun(A,$o),Nb)))),Nb) ) ) ).

% card_roots_unity
tff(fact_4870_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N3),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),N3)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_4871_card__sum__le__nat__sum,axiom,
    ! [S3: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ca(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S3)))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ca(nat,nat)),S3)) ).

% card_sum_le_nat_sum
tff(fact_4872_card__nth__roots,axiom,
    ! [C3: complex,Nb: nat] :
      ( ( C3 != 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_jg(complex,fun(nat,fun(complex,$o)),C3),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_4873_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,$o),set(complex),collect(complex),aTP_Lamp_bu(nat,fun(complex,$o),Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_4874_card__2__iff,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S3) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X3: A,Y5: A] :
          ( ( S3 = aa(set(A),set(A),insert(A,X3),aa(set(A),set(A),insert(A,Y5),bot_bot(set(A)))) )
          & ( X3 != Y5 ) ) ) ).

% card_2_iff
tff(fact_4875_card__3__iff,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S3) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X3: A,Y5: A,Z2: A] :
          ( ( S3 = aa(set(A),set(A),insert(A,X3),aa(set(A),set(A),insert(A,Y5),aa(set(A),set(A),insert(A,Z2),bot_bot(set(A))))) )
          & ( X3 != Y5 )
          & ( Y5 != Z2 )
          & ( X3 != Z2 ) ) ) ).

% card_3_iff
tff(fact_4876_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(set(A),nat,finite_card(A),A5))
     => ( A5 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_4877_card__Suc__Diff1,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( member(A,Xb,A5)
       => ( 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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_Suc_Diff1
tff(fact_4878_card_Oinsert__remove,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert(A,Xb),A5)) = 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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_4879_card_Oremove,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( member(A,Xb,A5)
       => ( aa(set(A),nat,finite_card(A),A5) = 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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_4880_card__Diff1__less,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( member(A,Xb,A5)
       => 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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)) ) ) ).

% card_Diff1_less
tff(fact_4881_card__Diff2__less,axiom,
    ! [A: $tType,A5: set(A),Xb: A,Y: A] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( member(A,Xb,A5)
       => ( member(A,Y,A5)
         => 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)),A5),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),A5)) ) ) ) ).

% card_Diff2_less
tff(fact_4882_card__Diff1__less__iff,axiom,
    ! [A: $tType,A5: set(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5))
    <=> ( aa(set(A),$o,finite_finite(A),A5)
        & member(A,Xb,A5) ) ) ).

% card_Diff1_less_iff
tff(fact_4883_card__Diff__singleton__if,axiom,
    ! [A: $tType,A5: 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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))) = $ite(member(A,Xb,A5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),one_one(nat)),aa(set(A),nat,finite_card(A),A5)) ).

% card_Diff_singleton_if
tff(fact_4884_card__Diff__singleton,axiom,
    ! [A: $tType,Xb: A,A5: set(A)] :
      ( member(A,Xb,A5)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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),A5)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_4885_sum__norm__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S3: set(A),F3: fun(A,B),K5: real] :
          ( ! [X4: A] :
              ( member(A,X4,S3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X4))),K5) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),S3))),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),S3))),K5)) ) ) ).

% sum_norm_bound
tff(fact_4886_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B),Nb: B,K2: nat] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),Nb) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),K2)
           => ( aa(B,$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),F3),A5)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Nb),K2)) ) ) ) ) ).

% prod_le_power
tff(fact_4887_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A5: set(A),F3: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,I2)),K5) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),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),A5))),K5)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_4888_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A5: set(A),F3: fun(A,B),K5: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),divide_divide(B,K5,aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A5)))) )
         => ( aa(set(A),$o,finite_finite(A),A5)
           => ( ( A5 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5)),K5) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_4889_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_4890_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K2: nat,Nb: nat] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => aa(nat,$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,$o),aTP_Lamp_gl(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb)))),Nb) ) ) ) ).

% polyfun_roots_card
tff(fact_4891_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A2: A,B2: fun(A,B),C3: B] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( 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_le(A,fun(fun(A,B),fun(B,fun(A,B))),A2),B2),C3)),S3) = $ite(member(A,A2,S3),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),C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),S3)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C3),aa(set(A),nat,finite_card(A),S3))) ) ) ) ).

% prod_gen_delta
tff(fact_4892_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K2: nat,Nb: nat] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
           => ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_gl(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb)))
              & aa(nat,$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,$o),aTP_Lamp_gl(fun(nat,A),fun(nat,fun(A,$o)),C3),Nb)))),Nb) ) ) ) ) ).

% polyfun_rootbound
tff(fact_4893_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,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      (A4),
                      aa(nat,option(nat),some(nat),zero_zero(nat)),
                      $ite((B4),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [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_4894_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)) ) )
         => ( ! [Uv: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv)) )
               => ( ~ (Y)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu: $o] :
                  ( ( Xb = vEBT_Leaf((Uu),$true) )
                 => ( ~ (Y)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu),$true)) ) )
             => ( ! [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2) )
                   => ( (Y)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw,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_4895_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A5: set(A),K2: nat] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),aa(set(A),nat,finite_card(A),A5))
       => ( 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_lf(set(A),fun(nat,fun(list(A),$o)),A5),K2))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ca(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),A5)),K2)),one_one(nat)),aa(set(A),nat,finite_card(A),A5))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_4896_distinct__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$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,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_4897_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => aa(set(list(A)),$o,finite_finite(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_lf(set(A),fun(nat,fun(list(A),$o)),A5),Nb))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4898_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => distinct(A,Xs) ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_4899_distinct__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( distinct(A,Xs)
     => ( distinct(B,Ys)
       => distinct(product_prod(A,B),product(A,B,Xs,Ys)) ) ) ).

% distinct_product
tff(fact_4900_finite__distinct__list,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A5 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_4901_subseqs__distinctD,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))
     => ( distinct(A,Xs)
       => distinct(A,Ys) ) ) ).

% subseqs_distinctD
tff(fact_4902_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( I3 != J3 )
               => ( aa(nat,A,nth(A,Xs),I3) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_4903_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I: nat,J: nat] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$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),I) = aa(nat,A,nth(A,Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_4904_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_4905_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_4906_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ? [X4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X4) = Xb )
            & ! [Y4: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y4),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y4) = Xb ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4907_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A5: set(nat),B5: set(A)] :
      ( distinct(A,Xs)
     => ( ( A5 = set_ord_lessThan(nat,aa(list(A),nat,size_size(list(A)),Xs)) )
       => ( ( B5 = aa(list(A),set(A),set2(A),Xs) )
         => bij_betw(nat,A,nth(A,Xs),A5,B5) ) ) ) ).

% bij_betw_nth
tff(fact_4908_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ 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),I)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs,I,A2)) ) ) ).

% distinct_list_update
tff(fact_4909_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_4910_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ! [Uv: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv)) )
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv))) )
         => ( ! [Uu: $o] :
                ( ( Xb = vEBT_Leaf((Uu),$true) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu),$true)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va2: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( 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_4911_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)) )
         => ~ ! [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_4912_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K2: nat,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(set(A),nat,finite_card(A),A5))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_lg(nat,fun(set(A),fun(list(A),$o)),K2),A5))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_ca(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),A5)),K2)),one_one(nat)),aa(set(A),nat,finite_card(A),A5))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_4913_distinct__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,union(A,Xs,Ys))
    <=> distinct(A,Ys) ) ).

% distinct_union
tff(fact_4914_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,A),V: num,Nb: nat] : case_nat(A,A2,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),Nb)) ).

% case_nat_add_eq_if
tff(fact_4915_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,fun(A,A)),V: num,Nb: nat] :
      aa(nat,A,rec_nat(A,A2,F3),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),F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb)),aa(nat,A,rec_nat(A,A2,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Nb))) ) ).

% rec_nat_add_eq_if
tff(fact_4916_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_4917_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_4918_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,A),V: num] : case_nat(A,A2,F3,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F3,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_4919_rec__nat__numeral,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,fun(A,A)),V: num] :
      aa(nat,A,rec_nat(A,A2,F3),aa(num,nat,numeral_numeral(nat),V)) = $let(
        pv: nat,
        pv:= pred_numeral(V),
        aa(A,A,aa(nat,fun(A,A),F3,pv),aa(nat,A,rec_nat(A,A2,F3),pv)) ) ).

% rec_nat_numeral
tff(fact_4920_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_lh(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_4921_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X2: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X2)) = aa(nat,A,F22,X2) ).

% old.nat.simps(5)
tff(fact_4922_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_4923_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_li(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_4924_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_lj(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_4925_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_4926_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_lk(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc2
tff(fact_4927_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_ll(nat,fun(nat,nat),Nb),Mb) ).

% max_Suc1
tff(fact_4928_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_ca(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)) ).

% diff_Suc
tff(fact_4929_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_4930_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_4931_old_Orec__nat__def,axiom,
    ! [A: $tType,X: A,Xa: fun(nat,fun(A,A)),Xb2: nat] : aa(nat,A,rec_nat(A,X,Xa),Xb2) = the(A,rec_set_nat(A,X,Xa,Xb2)) ).

% old.rec_nat_def
tff(fact_4932_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xb: A,F3: fun(nat,A),Nb: nat] :
      case_nat(A,Xb,F3,Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_4933_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F3: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),Nb: nat] :
      ( ( F3 = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F3,aa(nat,nat,suc,Nb)) = aa(A,A,aa(nat,fun(A,A),F22,Nb),aa(nat,A,F3,Nb)) ) ) ).

% rec_nat_Suc_imp
tff(fact_4934_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_4935_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_4936_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,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)) ).

% bezw_0
tff(fact_4937_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_ca(nat,nat),Nat) ).

% pred_def
tff(fact_4938_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_4939_prod__decode__aux_Osimps,axiom,
    ! [K2: nat,Mb: nat] :
      nat_prod_decode_aux(K2,Mb) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),K2),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),K2),Mb)),nat_prod_decode_aux(aa(nat,nat,suc,K2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),aa(nat,nat,suc,K2)))) ).

% prod_decode_aux.simps
tff(fact_4940_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] : aa(int,int,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,K2)))) = aa(int,int,bit_se4197421643247451524op_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K2)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_4941_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_4942_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Mb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),A2) ) ).

% drop_bit_drop_bit
tff(fact_4943_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,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),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),B2)) ) ).

% drop_bit_and
tff(fact_4944_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,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),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),B2)) ) ).

% drop_bit_or
tff(fact_4945_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,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),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),B2)) ) ).

% drop_bit_xor
tff(fact_4946_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,B2: $o] :
          aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa($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_4947_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K2) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_4948_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K2)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K2),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_4949_drop__bit__minus__one,axiom,
    ! [Nb: nat] : aa(int,int,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_4950_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K2: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K2))) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(num,A,numeral_numeral(A),K2)) ) ).

% drop_bit_Suc_bit0
tff(fact_4951_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K2: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K2))) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(num,A,numeral_numeral(A),K2)) ) ).

% drop_bit_Suc_bit1
tff(fact_4952_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_4953_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K2: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K2))) = aa(A,A,bit_se4197421643247451524op_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K2)) ) ).

% drop_bit_numeral_bit0
tff(fact_4954_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K2: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K2))) = aa(A,A,bit_se4197421643247451524op_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K2)) ) ).

% drop_bit_numeral_bit1
tff(fact_4955_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K2: num] : aa(int,int,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,K2)))) = aa(int,int,bit_se4197421643247451524op_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))) ).

% drop_bit_Suc_minus_bit0
tff(fact_4956_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] : aa(int,int,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,K2)))) = aa(int,int,bit_se4197421643247451524op_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2))) ).

% drop_bit_numeral_minus_bit0
tff(fact_4957_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K2: num] : aa(int,int,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,K2)))) = aa(int,int,bit_se4197421643247451524op_bit(int,Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K2)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_4958_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,Mb),Nb)) = aa(A,A,bit_se4197421643247451524op_bit(A,Mb),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_drop_bit
tff(fact_4959_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,Mb: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),aa(nat,A,semiring_1_of_nat(A),Mb)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,Nb),Mb)) ) ).

% drop_bit_of_nat
tff(fact_4960_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 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_4961_drop__bit__push__bit__int,axiom,
    ! [Mb: nat,Nb: nat,K2: int] : aa(int,int,bit_se4197421643247451524op_bit(int,Mb),aa(int,int,bit_se4730199178511100633sh_bit(int,Nb),K2)) = aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),aa(int,int,bit_se4730199178511100633sh_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb)),K2)) ).

% drop_bit_push_bit_int
tff(fact_4962_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) = aa(A,A,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_4963_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,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)),aa(A,A,bit_se4197421643247451524op_bit(A,Mb),A2)) ) ).

% drop_bit_take_bit
tff(fact_4964_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : divide_divide(A,A2,aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),one_one(A))) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_4965_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),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_4966_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = A2 ) ).

% bits_ident
tff(fact_4967_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) = A2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_4968_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = divide_divide(A,aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% drop_bit_half
tff(fact_4969_drop__bit__int__def,axiom,
    ! [Nb: nat,K2: int] : aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K2) = divide_divide(int,K2,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_4970_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,A,bit_se4197421643247451524op_bit(A,Nb),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% drop_bit_Suc
tff(fact_4971_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) = 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_4972_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2))
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_4973_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A2),Nb)
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_4974_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,Mb: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,Nb),aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(A,A,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_4975_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2) = $ite(Nb = zero_zero(nat),A2,aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% drop_bit_rec
tff(fact_4976_Suc__0__div__numeral,axiom,
    ! [K2: num] : divide_divide(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K2)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K2)) ).

% Suc_0_div_numeral
tff(fact_4977_Suc__0__mod__numeral,axiom,
    ! [K2: num] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K2)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K2)) ).

% Suc_0_mod_numeral
tff(fact_4978_bit__cut__integer__code,axiom,
    ! [K2: code_integer] :
      code_bit_cut_integer(K2) = $ite(K2 = zero_zero(code_integer),aa($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,$o)),aTP_Lamp_lm(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K2)),code_divmod_abs(K2,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))))) ).

% bit_cut_integer_code
tff(fact_4979_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_4980_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K2: num,L: num] : divide_divide(A,aa(num,A,numeral_numeral(A),K2),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K2,L)) ) ).

% numeral_div_numeral
tff(fact_4981_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K2: num,L: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),K2),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K2,L)) ) ).

% numeral_mod_numeral
tff(fact_4982_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,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_4983_fst__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(Mb,Nb)) = divide_divide(nat,Mb,Nb) ).

% fst_divmod_nat
tff(fact_4984_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : 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_4985_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_4986_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,$o),P)) = the(product_prod(A,B),aTP_Lamp_ln(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% The_case_prod
tff(fact_4987_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_lo(B,fun(A,A))),Prod) ).

% snd_def
tff(fact_4988_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_lp(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_4989_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: 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),X),Xa) = aa(B,C,aa(A,fun(B,C),X,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_4990_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),X: 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),F3),X) = aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),X)),aa(product_prod(A,B),B,product_snd(A,B),X)) ).

% case_prod_beta'
tff(fact_4991_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,F3: fun(D,fun(B,C)),G3: fun(A,D)] : aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_lq(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),F3),G3) = 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_lr(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),F3),G3)) ).

% split_comp_eq
tff(fact_4992_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_4993_snd__conv,axiom,
    ! [B: $tType,A: $tType,X1: B,X2: 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),X2)) = X2 ).

% snd_conv
tff(fact_4994_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_4995_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_4996_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_4997_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X2: 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),X2)) = X1 ).

% fst_conv
tff(fact_4998_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_4999_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType,S2: product_prod(A,B),Ta: product_prod(A,B)] :
      ( ( S2 = Ta )
    <=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S2) = aa(product_prod(A,B),A,product_fst(A,B),Ta) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),S2) = aa(product_prod(A,B),B,product_snd(A,B),Ta) ) ) ) ).

% prod_eq_iff
tff(fact_5000_prod__eqI,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),Q3: product_prod(A,B)] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),P2) = aa(product_prod(A,B),A,product_fst(A,B),Q3) )
     => ( ( aa(product_prod(A,B),B,product_snd(A,B),P2) = aa(product_prod(A,B),B,product_snd(A,B),Q3) )
       => ( P2 = Q3 ) ) ) ).

% prod_eqI
tff(fact_5001_prod_Oexpand,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B),Prod2: product_prod(A,B)] :
      ( ( ( aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_fst(A,B),Prod2) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,product_snd(A,B),Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_5002_Product__Type_OCollect__case__prodD,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B),A5: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),Xb,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),A5)))
     => aa(B,$o,aa(A,fun(B,$o),A5,aa(product_prod(A,B),A,product_fst(A,B),Xb)),aa(product_prod(A,B),B,product_snd(A,B),Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_5003_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,A)),P2: 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),F3),P2) = aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),P2)),aa(product_prod(B,C),C,product_snd(B,C),P2)) ).

% case_prod_beta
tff(fact_5004_split__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,A)),Prod: 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),F3),Prod) = aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) ).

% split_beta
tff(fact_5005_prod_Osplit__sel__asm,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F3: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),Prod))
    <=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
          & ~ aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F3,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_5006_prod_Osplit__sel,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F3: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),Prod))
    <=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
       => aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F3,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_5007_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K2: int] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,Nb),aa(int,nat,nat2,K2)) = aa(int,nat,nat2,aa(int,int,bit_se4197421643247451524op_bit(int,Nb),K2)) ).

% drop_bit_nat_eq
tff(fact_5008_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)) = divide_divide(A,aa(num,A,numeral_numeral(A),Mb),aa(num,A,numeral_numeral(A),Nb)) ) ).

% fst_divmod
tff(fact_5009_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_5010_drop__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,Nb),Mb) = 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_5011_bit__cut__integer__def,axiom,
    ! [K2: code_integer] : code_bit_cut_integer(K2) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),divide_divide(code_integer,K2,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),~ dvd_dvd(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)),K2)) ).

% bit_cut_integer_def
tff(fact_5012_divmod__integer__code,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = 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)),K2),code_divmod_abs(K2,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_ls(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K2,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)),K2),
            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),K2),zero_zero(code_integer)),code_divmod_abs(K2,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_lt(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K2,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_5013_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),Nb: nat,Xs: list(A)] :
      ( member(product_prod(nat,A),P2,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,Nb,Xs)))
    <=> ( aa(nat,$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_5014_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_5015_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: 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),F3),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,F3,Y)) ).

% apsnd_conv
tff(fact_5016_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_5017_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xb: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),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),F3),Xb)) = aa(product_prod(A,C),A,product_fst(A,C),Xb) ).

% fst_apsnd
tff(fact_5018_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xb: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),aa(fun(C,A),fun(product_prod(B,C),product_prod(B,A)),product_apsnd(C,A,B),F3),Xb)) = aa(C,A,F3,aa(product_prod(B,C),C,product_snd(B,C),Xb)) ).

% snd_apsnd
tff(fact_5019_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xb: product_prod(A,C),G3: fun(C,B)] :
      ( ( 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),F3),Xb) = 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),G3),Xb) )
    <=> ( aa(C,B,F3,aa(product_prod(A,C),C,product_snd(A,C),Xb)) = aa(C,B,G3,aa(product_prod(A,C),C,product_snd(A,C),Xb)) ) ) ).

% apsnd_eq_conv
tff(fact_5020_distinct__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : distinct(product_prod(nat,A),enumerate(A,Nb,Xs)) ).

% distinct_enumerate
tff(fact_5021_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),divide_divide(nat,Xb,Y))))) ) ) ).

% bezw_non_0
tff(fact_5022_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),divide_divide(nat,Xb,Xaa)))))) ) ) ).

% bezw.elims
tff(fact_5023_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),divide_divide(nat,Xb,Y)))))) ).

% bezw.simps
tff(fact_5024_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P2)))),one_one(int)) ).

% rat_sgn_code
tff(fact_5025_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_5026_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),P2: A,Q: fun(B,$o),Q3: B] :
      ( aa(A,$o,P,P2)
     => ( aa(B,$o,Q,Q3)
       => ( aa(A,$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),Q3)))
          & 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),Q3))) ) ) ) ).

% conjI_realizer
tff(fact_5027_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_5028_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_5029_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),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_5030_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      adjust_mod(L,R2) = $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R2)) ).

% Divides.adjust_mod_def
tff(fact_5031_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_5032_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:= 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),divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P2),a2)),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),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),divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P2),a2)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P2),a2)) ) ) ) ).

% normalize_def
tff(fact_5033_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G3: fun(B,nat),P2: product_prod(A,B)] : basic_BNF_size_prod(A,B,F3,G3,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,F3,aa(product_prod(A,B),A,product_fst(A,B),P2))),aa(B,nat,G3,aa(product_prod(A,B),B,product_snd(A,B),P2)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_5034_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : gcd_gcd(A,one_one(A),A2) = one_one(A) ) ).

% gcd.bottom_left_bottom
tff(fact_5035_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : gcd_gcd(A,A2,one_one(A)) = one_one(A) ) ).

% gcd.bottom_right_bottom
tff(fact_5036_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,Nb: A] : gcd_gcd(A,Mb,aa(A,A,aa(A,fun(A,A),plus_plus(A),Mb),Nb)) = gcd_gcd(A,Mb,Nb) ) ).

% gcd_add2
tff(fact_5037_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,Nb: A] : gcd_gcd(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Mb),Nb),Nb) = gcd_gcd(A,Mb,Nb) ) ).

% gcd_add1
tff(fact_5038_gcd__exp,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A2: A,Nb: nat,B2: 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),gcd_gcd(A,A2,B2)),Nb) ) ).

% gcd_exp
tff(fact_5039_gcd__1__int,axiom,
    ! [Mb: int] : gcd_gcd(int,Mb,one_one(int)) = one_one(int) ).

% gcd_1_int
tff(fact_5040_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,Nb: num] : gcd_gcd(A,A2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = gcd_gcd(A,A2,aa(num,A,numeral_numeral(A),Nb)) ) ).

% gcd_neg_numeral_2
tff(fact_5041_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: num,A2: A] : gcd_gcd(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)),A2) = gcd_gcd(A,aa(num,A,numeral_numeral(A),Nb),A2) ) ).

% gcd_neg_numeral_1
tff(fact_5042_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,gcd_gcd(A,A2,B2),one_one(A))
        <=> ( gcd_gcd(A,A2,B2) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_5043_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: A,Mb: A] : gcd_gcd(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Nb),Mb),Nb) = gcd_gcd(A,Mb,Nb) ) ).

% gcd_diff2
tff(fact_5044_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Mb: A,Nb: A] : gcd_gcd(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Mb),Nb),Nb) = gcd_gcd(A,Mb,Nb) ) ).

% gcd_diff1
tff(fact_5045_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Mb: A,K2: A,Nb: 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),K2),Mb)),Nb)) = gcd_gcd(A,Mb,Nb) ) ).

% gcd_add_mult
tff(fact_5046_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,K2: A] : dvd_dvd(A,gcd_gcd(A,A2,B2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),B2)) ) ).

% gcd_dvd_prod
tff(fact_5047_bezout__int,axiom,
    ! [Xb: int,Y: int] :
    ? [U2: int,V3: 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),V3),Y)) = gcd_gcd(int,Xb,Y) ).

% bezout_int
tff(fact_5048_gcd__mult__distrib__int,axiom,
    ! [K2: int,Mb: int,Nb: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K2)),gcd_gcd(int,Mb,Nb)) = gcd_gcd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K2),Mb),aa(int,int,aa(int,fun(int,int),times_times(int),K2),Nb)) ).

% gcd_mult_distrib_int
tff(fact_5049_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C3) = gcd_gcd(A,B2,C3) ) ) ) ).

% gcd_mult_unit1
tff(fact_5050_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( gcd_gcd(A,B2,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)) = gcd_gcd(A,B2,C3) ) ) ) ).

% gcd_mult_unit2
tff(fact_5051_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( gcd_gcd(A,B2,divide_divide(A,C3,A2)) = gcd_gcd(A,B2,C3) ) ) ) ).

% gcd_div_unit2
tff(fact_5052_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C3: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( gcd_gcd(A,divide_divide(A,B2,A2),C3) = gcd_gcd(A,B2,C3) ) ) ) ).

% gcd_div_unit1
tff(fact_5053_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun(nat,$o),Mb: nat] :
      ( ! [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),K)
         => aa(nat,$o,P,K) )
     => ( ! [K: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),I4)
                 => aa(nat,$o,P,I4) )
             => aa(nat,$o,P,K) ) )
       => aa(nat,$o,P,Mb) ) ) ).

% nat_descend_induct
tff(fact_5054_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q3: product_prod(A,B),F3: fun(A,fun(B,C)),G3: fun(A,fun(B,C)),P2: product_prod(A,B)] :
      ( ! [X4: A,Y3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) = Q3 )
         => ( aa(B,C,aa(A,fun(B,C),F3,X4),Y3) = aa(B,C,aa(A,fun(B,C),G3,X4),Y3) ) )
     => ( ( P2 = Q3 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),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),G3),Q3) ) ) ) ).

% split_cong
tff(fact_5055_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,aa(list(A),list(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_5056_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_5057_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_5058_gcd__1__nat,axiom,
    ! [Mb: nat] : gcd_gcd(nat,Mb,one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_5059_gcd__Suc__0,axiom,
    ! [Mb: nat] : gcd_gcd(nat,Mb,aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_5060_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),gcd_gcd(nat,Mb,Nb))
    <=> ( ( Mb != zero_zero(nat) )
        | ( Nb != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_5061_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_5062_in__set__remove1,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( ( A2 != B2 )
     => ( member(A,A2,aa(list(A),set(A),set2(A),remove1(A,B2,Xs)))
      <=> member(A,A2,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% in_set_remove1
tff(fact_5063_set__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rotate1(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rotate1
tff(fact_5064_length__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate1(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate1
tff(fact_5065_distinct1__rotate,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),rotate1(A),Xs))
    <=> distinct(A,Xs) ) ).

% distinct1_rotate
tff(fact_5066_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K2)
         => ( set_or5935395276787703475ssThan(A,K2,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_5067_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_5068_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_5069_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite(A),set_or5935395276787703475ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioo_iff
tff(fact_5070_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))
     => ( aa(list(A),list(A),rotate1(A),Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5071_gcd__mult__distrib__nat,axiom,
    ! [K2: nat,Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),gcd_gcd(nat,Mb,Nb)) = gcd_gcd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Mb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K2),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_5072_notin__set__remove1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ~ member(A,Xb,aa(list(A),set(A),set2(A),remove1(A,Y,Xs))) ) ).

% notin_set_remove1
tff(fact_5073_remove1__idem,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( remove1(A,Xb,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_5074_remove1__commute,axiom,
    ! [A: $tType,Xb: A,Y: A,Zs: list(A)] : remove1(A,Xb,remove1(A,Y,Zs)) = remove1(A,Y,remove1(A,Xb,Zs)) ).

% remove1_commute
tff(fact_5075_distinct__remove1,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => distinct(A,remove1(A,Xb,Xs)) ) ).

% distinct_remove1
tff(fact_5076_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),gcd_gcd(nat,A2,B2)),B2) ) ).

% gcd_le2_nat
tff(fact_5077_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),gcd_gcd(nat,A2,B2)),A2) ) ).

% gcd_le1_nat
tff(fact_5078_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( gcd_gcd(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),Nb) = gcd_gcd(nat,Mb,Nb) ) ) ).

% gcd_diff1_nat
tff(fact_5079_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( gcd_gcd(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb),Nb) = gcd_gcd(nat,Mb,Nb) ) ) ).

% gcd_diff2_nat
tff(fact_5080_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite(A),set_or5935395276787703475ssThan(A,A2,B2)) ) ) ).

% infinite_Ioo
tff(fact_5081_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_5082_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X4: nat,Y3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),gcd_gcd(nat,A2,B2)) ) ).

% bezout_nat
tff(fact_5083_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X4: nat,Y3: 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),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = 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),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)) = gcd_gcd(nat,A2,B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_5084_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_5085_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_5086_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_5087_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_5088_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_5089_length__remove1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      aa(list(A),nat,size_size(list(A)),remove1(A,Xb,Xs)) = $ite(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_5090_bezw__aux,axiom,
    ! [Xb: nat,Y: nat] : aa(nat,int,semiring_1_of_nat(int),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_5091_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: 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,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_5092_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A5: set(C),F3: fun(C,A),G3: fun(C,B)] : bNF_Greatest_image2(C,A,B,A5,F3,G3) = aa(fun(product_prod(A,B),$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_lu(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),A5),F3),G3)) ).

% image2_def
tff(fact_5093_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite(nat),S3)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N4)
          & member(nat,N4,S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_5094_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_5095_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_5096_tanh__real__bounds,axiom,
    ! [Xb: real] : 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_5097_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F3: fun(B,A),Xb: B,C3: C,G3: fun(B,C),A5: set(B)] :
      ( ( B2 = aa(B,A,F3,Xb) )
     => ( ( C3 = aa(B,C,G3,Xb) )
       => ( member(B,Xb,A5)
         => member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C3),bNF_Greatest_image2(B,A,C,A5,F3,G3)) ) ) ) ).

% image2_eqI
tff(fact_5098_infinite__nat__iff__unbounded,axiom,
    ! [S3: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite(nat),S3)
    <=> ! [M5: nat] :
        ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N4)
          & member(nat,N4,S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_5099_unbounded__k__infinite,axiom,
    ! [K2: nat,S3: set(nat)] :
      ( ! [M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),M)
         => ? [N8: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),N8)
              & member(nat,N8,S3) ) )
     => ~ aa(set(nat),$o,finite_finite(nat),S3) ) ).

% unbounded_k_infinite
tff(fact_5100_finite__enumerate,axiom,
    ! [S3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),S3)
     => ? [R4: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R4,set_ord_lessThan(nat,aa(set(nat),nat,finite_card(nat),S3)))
          & ! [N8: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N8),aa(set(nat),nat,finite_card(nat),S3))
             => member(nat,aa(nat,nat,R4,N8),S3) ) ) ) ).

% finite_enumerate
tff(fact_5101_xor__minus__numerals_I1_J,axiom,
    ! [Nb: num,K2: 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))),K2) = 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)),K2)) ).

% xor_minus_numerals(1)
tff(fact_5102_xor__minus__numerals_I2_J,axiom,
    ! [K2: int,Nb: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K2),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),K2),neg_numeral_sub(int,Nb,one2))) ).

% xor_minus_numerals(2)
tff(fact_5103_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_5104_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_5105_sub__num__simps_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,aa(num,num,bit0,K2),aa(num,num,bit0,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K2,L)) ) ).

% sub_num_simps(6)
tff(fact_5106_sub__num__simps_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K2),aa(num,num,bit1,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K2,L)) ) ).

% sub_num_simps(9)
tff(fact_5107_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_5108_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_5109_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_5110_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_5111_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_5112_sub__num__simps_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K2),aa(num,num,bit0,L)) = neg_numeral_dbl_inc(A,neg_numeral_sub(A,K2,L)) ) ).

% sub_num_simps(8)
tff(fact_5113_sub__num__simps_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,aa(num,num,bit0,K2),aa(num,num,bit1,L)) = neg_numeral_dbl_dec(A,neg_numeral_sub(A,K2,L)) ) ).

% sub_num_simps(7)
tff(fact_5114_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_5115_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_5116_sub__num__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_sub(A,aa(num,num,bit1,K2),one2) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,K2)) ) ).

% sub_num_simps(5)
tff(fact_5117_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_5118_sub__num__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num] : neg_numeral_sub(A,aa(num,num,bit0,K2),one2) = aa(num,A,numeral_numeral(A),bitM(K2)) ) ).

% sub_num_simps(4)
tff(fact_5119_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_5120_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_5121_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_5122_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_5123_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_5124_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_5125_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_5126_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_5127_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_5128_neg__numeral__class_Osub__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K2: num,L: num] : neg_numeral_sub(A,K2,L) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),K2)),aa(num,A,numeral_numeral(A),L)) ) ).

% neg_numeral_class.sub_def
tff(fact_5129_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_5130_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_5131_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_5132_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_5133_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_5134_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_5135_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_5136_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( strict_mono_on(A,B,F3,A5)
        <=> ! [R5: A,S6: A] :
              ( ( member(A,R5,A5)
                & member(A,S6,A5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),R5),S6) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,R5)),aa(A,B,F3,S6)) ) ) ) ).

% strict_mono_on_def
tff(fact_5137_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [R4: A,S: A] :
              ( member(A,R4,A5)
             => ( member(A,S,A5)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R4),S)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,R4)),aa(A,B,F3,S)) ) ) )
         => strict_mono_on(A,B,F3,A5) ) ) ).

% strict_mono_onI
tff(fact_5138_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A5: set(A),R2: A,S2: A] :
          ( strict_mono_on(A,B,F3,A5)
         => ( member(A,R2,A5)
           => ( member(A,S2,A5)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R2),S2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,R2)),aa(A,B,F3,S2)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_5139_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F3: fun(A,B),A5: set(A),Xb: A,Y: A] :
          ( strict_mono_on(A,B,F3,A5)
         => ( member(A,Xb,A5)
           => ( member(A,Y,A5)
             => ( aa(A,$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,F3,Xb)),aa(A,B,F3,Y)) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_5140_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) )
           => ( 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),divide_divide(B,A2,B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_5141_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) )
           => ( 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),divide_divide(B,A2,B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_5142_bounded__linear__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( ? [K8: real] :
            ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K8))
         => real_V4916620083959148203axioms(A,B,F3) ) ) ).

% bounded_linear_axioms.intro
tff(fact_5143_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,Xb,Y),A2)
         => ( 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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,A2,Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_5144_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,Xb,Y),A2)
         => ( 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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,A2,Y)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_5145_distrib__left__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,A2: B,B2: B,C3: B] :
          ( nO_MATCH(A,B,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),C3)) = 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),C3)) ) ) ) ).

% distrib_left_NO_MATCH
tff(fact_5146_distrib__right__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring(B)
     => ! [Xb: A,Y: A,C3: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),C3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),C3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C3)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C3)) ) ) ) ).

% distrib_right_NO_MATCH
tff(fact_5147_left__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,C3: B,A2: B,B2: B] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),C3)
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),C3) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),A2),C3)),aa(B,B,aa(B,fun(B,B),times_times(B),B2),C3)) ) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_5148_right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType,B: $tType] :
      ( ring(B)
     => ! [Xb: A,Y: A,A2: B,B2: B,C3: B] :
          ( nO_MATCH(A,B,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),C3)) = 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),C3)) ) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_5149_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_5150_scale__left__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,C3: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),C3)
         => ( 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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_5151_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,Y: A,C3: B,A2: real,B2: real] :
          ( nO_MATCH(A,B,divide_divide(A,Xb,Y),C3)
         => ( 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),real_V8093663219630862766scaleR(A,A2,Xb)),real_V8093663219630862766scaleR(A,B2,Xb)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_5152_bounded__linear__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V4916620083959148203axioms(A,B,F3)
        <=> ? [K6: real] :
            ! [X3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K6)) ) ) ).

% bounded_linear_axioms_def
tff(fact_5153_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,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_lv(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_5154_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F3: fun(C,A),G3: fun(D,B),Xb: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),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),G3),Xb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,aa(product_prod(C,D),C,product_fst(C,D),Xb))),aa(D,B,G3,aa(product_prod(C,D),D,product_snd(C,D),Xb))) ).

% apfst_apsnd
tff(fact_5155_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: 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),F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),Xb)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G3,aa(product_prod(D,C),D,product_fst(D,C),Xb))),aa(C,B,F3,aa(product_prod(D,C),C,product_snd(D,C),Xb))) ).

% apsnd_apfst
tff(fact_5156_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_5157_funpow__0,axiom,
    ! [A: $tType,F3: 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)),F3),Xb) = Xb ).

% funpow_0
tff(fact_5158_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F3: fun(C,A),Xb: C,Y: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),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,F3,Xb)),Y) ).

% apfst_conv
tff(fact_5159_fst__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),Xb: product_prod(C,B)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),Xb)) = aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),Xb)) ).

% fst_apfst
tff(fact_5160_apfst__eq__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,A),Xb: product_prod(C,B),G3: fun(C,A)] :
      ( ( aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),Xb) = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,G3),Xb) )
    <=> ( aa(C,A,F3,aa(product_prod(C,B),C,product_fst(C,B),Xb)) = aa(C,A,G3,aa(product_prod(C,B),C,product_fst(C,B),Xb)) ) ) ).

% apfst_eq_conv
tff(fact_5161_snd__apfst,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,B),Xb: product_prod(C,A)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,A),product_prod(B,A),product_apfst(C,B,A,F3),Xb)) = aa(product_prod(C,A),A,product_snd(C,A),Xb) ).

% snd_apfst
tff(fact_5162_apsnd__apfst__commute,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F3: fun(C,B),G3: fun(D,A),P2: 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),F3),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G3),P2)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,G3),aa(product_prod(D,C),product_prod(D,B),aa(fun(C,B),fun(product_prod(D,C),product_prod(D,B)),product_apsnd(C,B,D),F3),P2)) ).

% apsnd_apfst_commute
tff(fact_5163_funpow__mod__eq,axiom,
    ! [A: $tType,Nb: nat,F3: 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),F3),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)),F3),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),F3),Xb) ) ) ).

% funpow_mod_eq
tff(fact_5164_funpow__swap1,axiom,
    ! [A: $tType,F3: fun(A,A),Nb: nat,Xb: A] : aa(A,A,F3,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),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),F3),aa(A,A,F3,Xb)) ).

% funpow_swap1
tff(fact_5165_funpow__mult,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,F3: 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),F3)) = 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)),F3) ).

% funpow_mult
tff(fact_5166_bij__betw__funpow,axiom,
    ! [A: $tType,F3: fun(A,A),S3: set(A),Nb: nat] :
      ( bij_betw(A,A,F3,S3,S3)
     => 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),F3),S3,S3) ) ).

% bij_betw_funpow
tff(fact_5167_funpow__times__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [F3: 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,F3,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,F3,Xb))) ) ).

% funpow_times_power
tff(fact_5168_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K2: num,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K2)),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),K2)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A2) ) ).

% numeral_add_unfold_funpow
tff(fact_5169_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_5170_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K2: num] : aa(num,A,numeral_numeral(A),K2) = 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),K2)),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% numeral_unfold_funpow
tff(fact_5171_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_5172_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,$o))),Nb),P),Xb),Y)
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = Xb )
          & ( aa(nat,A,F5,Nb) = Y )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
             => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,F5,I3)),aa(nat,A,F5,aa(nat,nat,suc,I3))) ) ) ) ).

% relpowp_fun_conv
tff(fact_5173_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_5174_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_5175_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,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z)
     => ~ ! [Y3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y3)
           => ~ aa(A,$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),Y3),Z) ) ) ).

% relpowp_Suc_E2
tff(fact_5176_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,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z)
     => ? [Y3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y3)
          & aa(A,$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),Y3),Z) ) ) ).

% relpowp_Suc_D2
tff(fact_5177_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_5178_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,$o))),aa(nat,nat,suc,Nb)),P),Xb),Z)
     => ~ ! [Y3: A] :
            ( aa(A,$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),Y3)
           => ~ aa(A,$o,aa(A,fun(A,$o),P,Y3),Z) ) ) ).

% relpowp_Suc_E
tff(fact_5179_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,$o))),Nb),P),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(A,$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),Y3)
               => ~ aa(A,$o,aa(A,fun(A,$o),P,Y3),Z) ) ) ) ) ).

% relpowp_E
tff(fact_5180_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,$o))),Nb),P),Xb),Z)
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( aa(A,$o,aa(A,fun(A,$o),P,Xb),Y3)
               => ~ aa(A,$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),Y3),Z) ) ) ) ) ).

% relpowp_E2
tff(fact_5181_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_5182_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q3: product_prod(A,B),F3: fun(C,A),P2: product_prod(C,B)] :
      ( ( Q3 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),P2) )
     => ~ ! [X4: C,Y3: B] :
            ( ( P2 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X4),Y3) )
           => ( Q3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F3,X4)),Y3) ) ) ) ).

% apfst_convE
tff(fact_5183_divmod__integer__eq__cases,axiom,
    ! [K2: code_integer,L: code_integer] :
      code_divmod_integer(K2,L) = $ite(
        K2 = 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)),K2),
          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),K2) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K2,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_lw(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K2,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_5184_comp__funpow,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F3: 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,F3)) = 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),F3)) ).

% comp_funpow
tff(fact_5185_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(C,B),G3: fun(D,C),Xb: product_prod(A,D)] : 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),F3),aa(product_prod(A,D),product_prod(A,C),aa(fun(D,C),fun(product_prod(A,D),product_prod(A,C)),product_apsnd(D,C,A),G3),Xb)) = aa(product_prod(A,D),product_prod(A,B),aa(fun(D,B),fun(product_prod(A,D),product_prod(A,B)),product_apsnd(D,B,A),aa(fun(D,C),fun(D,B),comp(C,B,D,F3),G3)),Xb) ).

% apsnd_compose
tff(fact_5186_apfst__compose,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,C),Xb: product_prod(D,B)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F3),aa(product_prod(D,B),product_prod(C,B),product_apfst(D,C,B,G3),Xb)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,aa(fun(D,C),fun(D,A),comp(C,A,D,F3),G3)),Xb) ).

% apfst_compose
tff(fact_5187_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,F3: 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)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3)) ).

% funpow.simps(2)
tff(fact_5188_funpow__Suc__right,axiom,
    ! [A: $tType,Nb: nat,F3: 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)),F3) = 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),F3)),F3) ).

% funpow_Suc_right
tff(fact_5189_funpow__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,F3: 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)),F3) = 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),F3)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3)) ).

% funpow_add
tff(fact_5190_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G3: fun(C,B),A5: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X4: B,Y3: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X4)),aa(B,A,H,Y3))
           => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,H),G3)),A5) = aa(B,A,H,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G3),A5)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_5191_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(D,fun(C,A)),G3: 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,F3),G3)),Xb) = aa(C,A,aa(D,fun(C,A),F3,aa(B,D,G3,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_5192_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ah(nat,fun(nat,$o)),aTP_Lamp_ai(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_5193_set__removeAll,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(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_5194_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_5195_removeAll__id,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(list(A),list(A),removeAll(A,Xb),Xs) = Xs ) ) ).

% removeAll_id
tff(fact_5196_fst__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C),comp(product_prod(C,B),C,product_prod(A,B),product_fst(C,B)),product_apfst(A,C,B,F3)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)) ).

% fst_comp_apfst
tff(fact_5197_snd__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B),comp(product_prod(C,B),B,product_prod(A,B),product_snd(C,B)),product_apfst(A,C,B,F3)) = product_snd(A,B) ).

% snd_comp_apfst
tff(fact_5198_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),comp(product_prod(A,C),A,product_prod(A,B),product_fst(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = product_fst(A,B) ).

% fst_comp_apsnd
tff(fact_5199_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),comp(product_prod(A,C),C,product_prod(A,B),product_snd(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),F3),product_snd(A,B)) ).

% snd_comp_apsnd
tff(fact_5200_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_lx(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_5201_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_ly(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_5202_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_lx(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_5203_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_ly(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_5204_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_5205_distinct__removeAll,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),removeAll(A,Xb),Xs)) ) ).

% distinct_removeAll
tff(fact_5206_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)),aa(list(A),list(A),removeAll(A,Xb),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_removeAll_less_eq
tff(fact_5207_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),suc)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_5208_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3),suc)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_5209_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_5210_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_5211_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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,G3),suc)),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_5212_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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,G3),suc)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_5213_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = 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,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_5214_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,K2: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),K2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2))) = 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,G3),aa(nat,fun(nat,nat),plus_plus(nat),K2))),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_5215_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4197421643247451524op_bit(A,Nb),A2)) = aa(fun(nat,nat),fun(nat,$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_5216_distinct__remove1__removeAll,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( remove1(A,Xb,Xs) = aa(list(A),list(A),removeAll(A,Xb),Xs) ) ) ).

% distinct_remove1_removeAll
tff(fact_5217_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),comp(A,A,nat,inverse_inverse(A)),F3))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_lz(fun(nat,A),fun(A,fun(nat,A)),F3),C3)) ) ) ).

% summable_inverse_divide
tff(fact_5218_length__removeAll__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,Xb),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_removeAll_less
tff(fact_5219_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_5220_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),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,G3,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_5221_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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,G3,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,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_5222_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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,G3,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,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_5223_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),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_5224_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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,G3),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_5225_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: 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,G3),aTP_Lamp_ma(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_5226_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: 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,G3),aTP_Lamp_ma(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_5227_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: 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,G3),aTP_Lamp_ma(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),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_5228_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: 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,G3),aTP_Lamp_ma(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),G3),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_5229_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),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_5230_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$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),G3),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,G3),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_5231_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_mb(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_5232_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_mc(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_5233_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S3: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ~ ? [X: A] :
                  ( member(A,X,S3)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S3))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_5234_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_5235_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_5236_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S3: set(A),Y: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( member(A,Y,S3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S3))),aa(A,B,F3,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_5237_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_me(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_5238_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) )
    <=> ? [A6: B] : P2 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A6),B2) ) ).

% eq_snd_iff
tff(fact_5239_Gcd__remove0__nat,axiom,
    ! [M6: set(nat)] :
      ( aa(set(nat),$o,finite_finite(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_5240_Gcd__1,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( member(A,one_one(A),A5)
         => ( gcd_Gcd(A,A5) = one_one(A) ) ) ) ).

% Gcd_1
tff(fact_5241_Gcd__nat__eq__one,axiom,
    ! [N3: set(nat)] :
      ( member(nat,one_one(nat),N3)
     => ( gcd_Gcd(nat,N3) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_5242_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y3: 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),X4),Y3)) ).

% eq_Abs_Integ
tff(fact_5243_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A5: set(A)] :
          ( dvd_dvd(A,A2,one_one(A))
         => ( member(A,A2,A5)
           => ( gcd_Gcd(A,A5) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_5244_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_5245_sum_Osize__neq,axiom,
    ! [A: $tType,B: $tType,Xb: sum_sum(A,B)] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),Xb) != zero_zero(nat) ).

% sum.size_neq
tff(fact_5246_prod_Osize__neq,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B)] : aa(product_prod(A,B),nat,size_size(product_prod(A,B)),Xb) != zero_zero(nat) ).

% prod.size_neq
tff(fact_5247_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_5248_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_5249_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_mf(nat,fun(nat,product_prod(nat,nat)))),Xb)) ).

% uminus_int.abs_eq
tff(fact_5250_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_5251_of__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: product_prod(nat,nat)] : aa(int,A,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_mg(nat,fun(nat,A))),Xb) ) ).

% of_int.abs_eq
tff(fact_5252_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),$o)),aTP_Lamp_mi(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_5253_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),$o)),aTP_Lamp_mk(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_5254_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_mm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_5255_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_mo(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_5256_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) )
    <=> ? [B6: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B6) ) ).

% eq_fst_iff
tff(fact_5257_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_5258_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_5259_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_5260_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q3)) = Q3 ).

% num_of_nat_numeral_eq
tff(fact_5261_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_5262_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_5263_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_5264_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_5265_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_5266_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_5267_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_5268_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_5269_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_5270_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_5271_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_5272_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_5273_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_5274_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_5275_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_5276_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_5277_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_5278_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_5279_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_5280_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_5281_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_5282_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_5283_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_5284_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_5285_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_5286_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),$o)),aTP_Lamp_mk(nat,fun(nat,fun(product_prod(nat,nat),$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_5287_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),$o)),aTP_Lamp_mi(nat,fun(nat,fun(product_prod(nat,nat),$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_5288_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_mp(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_5289_prod__encode__eq,axiom,
    ! [Xb: product_prod(nat,nat),Y: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),nat,nat_prod_encode,Xb) = aa(product_prod(nat,nat),nat,nat_prod_encode,Y) )
    <=> ( Xb = Y ) ) ).

% prod_encode_eq
tff(fact_5290_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_5291_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_5292_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_5293_of__int_Orep__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: int] : aa(int,A,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_mg(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,Xb)) ) ).

% of_int.rep_eq
tff(fact_5294_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,Mb: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K2,Mb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K2)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_5295_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_mf(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_5296_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),P2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ar(set(A),fun(fun(A,B),fun(A,$o)),I6),P2)))
         => ( groups1962203154675924110t_prod(A,B,P2,aa(set(A),set(A),insert(A,I),I6)) = $ite(member(A,I,I6),groups1962203154675924110t_prod(A,B,P2,I6),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P2,I)),groups1962203154675924110t_prod(A,B,P2,I6))) ) ) ) ).

% prod.insert'
tff(fact_5297_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))) = remove1(A,Xb,aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5298_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(list(A),set(A),set2(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) = A5 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_5299_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_5300_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(list(A),nat,size_size(list(A)),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) = aa(set(A),nat,finite_card(A),A5) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_5301_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),I6: set(B)] : groups1962203154675924110t_prod(B,A,G3,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_mq(fun(B,A),fun(set(B),fun(B,$o)),G3),I6))) = groups1962203154675924110t_prod(B,A,G3,I6) ) ).

% prod.non_neutral'
tff(fact_5302_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : distinct(A,aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_5303_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = aa(set(A),list(A),linord4507533701916653071of_set(A),B5) )
         => ( aa(set(A),$o,finite_finite(A),A5)
           => ( aa(set(A),$o,finite_finite(A),B5)
             => ( A5 = B5 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_5304_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),I6)
         => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mr(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H),I6) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G3,I6)),groups1962203154675924110t_prod(A,B,H,I6)) ) ) ) ).

% prod.distrib_triv'
tff(fact_5305_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
               => ( aa(A,B,G3,X4) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G3,S3) = groups1962203154675924110t_prod(A,B,G3,T2) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_5306_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
               => ( aa(A,B,G3,X4) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G3,T2) = groups1962203154675924110t_prod(A,B,G3,S3) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_5307_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T2: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
               => ( aa(A,B,H,I2) = one_one(B) ) )
           => ( ! [X4: A] :
                  ( member(A,X4,S3)
                 => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
             => ( groups1962203154675924110t_prod(A,B,G3,S3) = groups1962203154675924110t_prod(A,B,H,T2) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_5308_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),T2: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),T2)
         => ( ! [X4: A] :
                ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
               => ( aa(A,B,G3,X4) = one_one(B) ) )
           => ( ! [X4: A] :
                  ( member(A,X4,S3)
                 => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
             => ( groups1962203154675924110t_prod(A,B,G3,T2) = groups1962203154675924110t_prod(A,B,H,S3) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_5309_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I6: set(A),G3: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ar(set(A),fun(fun(A,B),fun(A,$o)),I6),G3)))
         => ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ar(set(A),fun(fun(A,B),fun(A,$o)),I6),H)))
           => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mr(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H),I6) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G3,I6)),groups1962203154675924110t_prod(A,B,H,I6)) ) ) ) ) ).

% prod.distrib'
tff(fact_5310_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: fun(B,A),I6: set(B)] :
          groups1962203154675924110t_prod(B,A,P2,I6) = $ite(aa(set(B),$o,finite_finite(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_mq(fun(B,A),fun(set(B),fun(B,$o)),P2),I6))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P2),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_mq(fun(B,A),fun(set(B),fun(B,$o)),P2),I6))),one_one(A)) ) ).

% prod.G_def
tff(fact_5311_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I: 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,I)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_5312_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_me(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_5313_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_mo(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_5314_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_mm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_5315_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),insert(A,Xb),A5)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),Xb),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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_5316_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I: 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),I))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I,J))),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5317_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_5318_remove1__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: A,F3: fun(A,B),Xs: list(A)] : remove1(A,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),Xs)) = Xs ) ).

% remove1_insort_key
tff(fact_5319_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_5320_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K2)
         => ( set_or3652927894154168847AtMost(A,K2,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_5321_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K2,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_5322_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K2: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K2,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_5323_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite(A),set_or3652927894154168847AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioc_iff
tff(fact_5324_length__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: 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,F3),Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_insort
tff(fact_5325_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_5326_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ~ member(A,Xb,A5)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),insert(A,Xb),A5)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),Xb),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_5327_insort__key__left__comm,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xb: B,Y: B,Xs: list(B)] :
          ( ( aa(B,A,F3,Xb) != aa(B,A,F3,Y) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Y),Xs)) ) ) ) ).

% insort_key_left_comm
tff(fact_5328_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C3,D3) )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C3) )
            | ( ( A2 = C3 )
              & ( B2 = D3 ) ) ) ) ) ).

% Ioc_inj
tff(fact_5329_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_ms(A,A)),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),Xb),Xs)) ) ).

% insort_left_comm
tff(fact_5330_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_ms(A,A)),Y)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(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_ms(A,A)),Xb)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_5331_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_5332_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_5333_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_5334_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_5335_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite(A),set_or3652927894154168847AtMost(A,A2,B2)) ) ) ).

% infinite_Ioc
tff(fact_5336_set__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: 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,F3),Xb),Xs)) = aa(set(A),set(A),insert(A,Xb),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_key
tff(fact_5337_distinct__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: 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,F3),Xb),Xs))
        <=> ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
            & distinct(A,Xs) ) ) ) ).

% distinct_insort
tff(fact_5338_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_5339_numeral__sqr,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K2: num] : aa(num,A,numeral_numeral(A),sqr(K2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K2)),aa(num,A,numeral_numeral(A),K2)) ) ).

% numeral_sqr
tff(fact_5340_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Mb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G3),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% sum.head
tff(fact_5341_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Mb: nat,Nb: nat,G3: fun(nat,A)] :
          ( aa(nat,$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),G3),set_or1337092689740270186AtMost(nat,Mb,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Mb)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or3652927894154168847AtMost(nat,Mb,Nb))) ) ) ) ).

% prod.head
tff(fact_5342_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_5343_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D3) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_5344_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C3,D3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_5345_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_5346_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_5347_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_5348_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),Xb),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5349_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_ms(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_5350_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( member(list(A),Xs,aa(list(list(A)),set(list(A)),set2(list(A)),product_lists(A,Xss)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Xss) ) ) ).

% in_set_product_lists_length
tff(fact_5351_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => ( (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:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X3,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X3: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),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,$o))))),aTP_Lamp_mt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5352_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_5353_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_mu(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_5354_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_mv(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_5355_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_mw(fun(B,A),fun(fun(C,B),fun(C,A)),H),F22),Option) ).

% option.case_distrib
tff(fact_5356_option_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),X2: B] : case_option(A,B,F1,F22,aa(B,option(B),some(B),X2)) = aa(B,A,F22,X2) ).

% option.simps(5)
tff(fact_5357_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_5358_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),Xb: option(A)] :
      ( case_option($o,A,(P),Q,Xb)
     => ( ( ( Xb = none(A) )
         => ~ (P) )
       => ~ ! [Y3: A] :
              ( ( Xb = aa(A,option(A),some(A),Y3) )
             => ~ aa(A,$o,Q,Y3) ) ) ) ).

% case_optionE
tff(fact_5359_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_5360_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_5361_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_5362_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_5363_distinct__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ! [X4: list(A)] :
          ( member(list(A),X4,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => distinct(A,X4) )
     => distinct(list(A),product_lists(A,Xss)) ) ).

% distinct_product_lists
tff(fact_5364_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:= divide_divide(nat,Dega,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
            $let(
              m2: nat,
              m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Dega),n),
              ( ! [X3: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                 => vEBT_VEBT_valid(X3,n) )
              & vEBT_VEBT_valid(Summarya,m2)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X7)
                  & ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),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,$o))))),aTP_Lamp_mt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Dega),TreeLista),Summarya),n),m2)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5365_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => ( 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:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                    $let(
                      m2: nat,
                      m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                      ( ! [X3: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_VEBT_valid(X3,n) )
                      & vEBT_VEBT_valid(Summary,m2)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                          & ! [X3: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),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,$o))))),aTP_Lamp_mt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5366_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => ( 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:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                      $let(
                        m2: nat,
                        m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                        ( ! [X3: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_VEBT_valid(X3,n) )
                        & vEBT_VEBT_valid(Summary,m2)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                            & ! [X3: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),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,$o))))),aTP_Lamp_mt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_5367_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))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( 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((Uu),(Uv))),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:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                        $let(
                          m2: nat,
                          m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                          ( ! [X3: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X3,n) )
                          & vEBT_VEBT_valid(Summary,m2)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X3: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),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,$o))))),aTP_Lamp_mt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_5368_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))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( 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((Uu),(Uv))),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:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X3,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X3: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),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,$o))))),aTP_Lamp_mt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_5369_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))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( ( (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((Uu),(Uv))),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:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),
                          $let(
                            m2: nat,
                            m2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n),
                            ( ! [X3: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X3,n) )
                            & vEBT_VEBT_valid(Summary,m2)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m2) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X3: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),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,$o))))),aTP_Lamp_mt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$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_5370_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_mx(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_5371_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_my(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_5372_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_5373_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_5374_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_5375_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_5376_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_mz(num,option(num)),bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_5377_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_5378_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit0,Mb)) = case_option(option(num),num,none(num),aTP_Lamp_mz(num,option(num)),bit_take_bit_num(pred_numeral(R2),Mb)) ).

% take_bit_num_simps(6)
tff(fact_5379_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit1,Mb)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R2),Mb))) ).

% take_bit_num_simps(7)
tff(fact_5380_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_5381_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_5382_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_5383_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_5384_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_5385_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_na(num,fun(nat,option(num)),Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_5386_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_nb(nat,option(num)),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_5387_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_5388_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_5389_GreatestI__nat,axiom,
    ! [P: fun(nat,$o),K2: nat,B2: nat] :
      ( aa(nat,$o,P,K2)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_nat
tff(fact_5390_Greatest__le__nat,axiom,
    ! [P: fun(nat,$o),K2: nat,B2: nat] :
      ( aa(nat,$o,P,K2)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),order_Greatest(nat,P)) ) ) ).

% Greatest_le_nat
tff(fact_5391_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),B2: nat] :
      ( ? [X_1: nat] : aa(nat,$o,P,X_1)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_ex_nat
tff(fact_5392_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_5393_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: num,Q3: num] :
          ( ( bit_take_bit_num(Mb,Nb) = aa(num,option(num),some(num),Q3) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Q3) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_5394_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_nc(num,fun(nat,option(num)),Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_5395_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
           => ( ! [X4: A] :
                  ( aa(A,$o,P,X4)
                 => ( ! [Y4: A] :
                        ( aa(A,$o,P,Y4)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X4) )
                   => aa(A,$o,Q,X4) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_5396_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y3: A] :
                ( aa(A,$o,P,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
           => ( order_Greatest(A,P) = Xb ) ) ) ) ).

% Greatest_equality
tff(fact_5397_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_5398_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_5399_and__not__num__eq__Some__iff,axiom,
    ! [Mb: num,Nb: num,Q3: num] :
      ( ( bit_and_not_num(Mb,Nb) = aa(num,option(num),some(num),Q3) )
    <=> ( 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),Q3) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_5400_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_nd(num,option(num)),bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(8)
tff(fact_5401_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_5402_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_5403_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_5404_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_5405_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_nh(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_5406_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys3: list(A)] :
            ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys3) )
       => ( ! [Ys3: list(A),Zs2: list(A)] :
              ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( member(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys3 != Zs2 )
                 => ( 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),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_5407_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I6)) = aa(fun(A,$o),set(A),collect(A),aa(set(nat),fun(A,$o),aTP_Lamp_ni(list(A),fun(set(nat),fun(A,$o)),Xs),I6)) ).

% set_nths
tff(fact_5408_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_5409_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))
        <=> ( aa(A,$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),C3) ) ) ) ).

% inf.bounded_iff
tff(fact_5410_Diff__disjoint,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_5411_Diff__Compl,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ).

% Diff_Compl
tff(fact_5412_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A5: set(A),F3: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_nj(fun(A,B),fun(fun(A,$o),fun(A,B)),F3),P)),A5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_5413_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A5: set(A),P: fun(A,$o),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_nk(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F3)),A5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_5414_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_5415_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_5416_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_5417_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_5418_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% inf.strict_boundedE
tff(fact_5419_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_5420_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.strict_coboundedI1
tff(fact_5421_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.strict_coboundedI2
tff(fact_5422_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.coboundedI2
tff(fact_5423_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.coboundedI1
tff(fact_5424_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_5425_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_5426_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_5427_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_5428_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_5429_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_5430_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)) ) ) ) ).

% inf.boundedI
tff(fact_5431_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))
         => ~ ( aa(A,$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),C3) ) ) ) ).

% inf.boundedE
tff(fact_5432_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_5433_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_5434_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_5435_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_5436_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_5437_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F3: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X4: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X4),Y3)),X4)
         => ( ! [X4: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X4),Y3)),Y3)
           => ( ! [X4: A,Y3: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F3,Xb),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_5438_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_5439_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_5440_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_5441_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_5442_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$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),C3),D3)) ) ) ) ).

% inf_mono
tff(fact_5443_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_5444_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_5445_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_5446_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_5447_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_5448_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_5449_Diff__Int__distrib2,axiom,
    ! [A: $tType,A5: set(A),B5: 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)),A5),B5)),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)),A5),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C5)) ).

% Diff_Int_distrib2
tff(fact_5450_Diff__Int__distrib,axiom,
    ! [A: $tType,C5: set(A),A5: set(A),B5: 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)),A5),B5)) = 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),A5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),B5)) ).

% Diff_Int_distrib
tff(fact_5451_Diff__Diff__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ).

% Diff_Diff_Int
tff(fact_5452_Diff__Int2,axiom,
    ! [A: $tType,A5: set(A),C5: set(A),B5: 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)),A5),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),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)),A5),C5)),B5) ).

% Diff_Int2
tff(fact_5453_Int__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: 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)),A5),B5)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C5)) ).

% Int_Diff
tff(fact_5454_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(num,B),F32: fun(num,B),Num: num] : aa(B,A,H,case_num(B,F1,F22,F32,Num)) = case_num(A,aa(B,A,H,F1),aa(fun(num,B),fun(num,A),aTP_Lamp_nl(fun(B,A),fun(fun(num,B),fun(num,A)),H),F22),aa(fun(num,B),fun(num,A),aTP_Lamp_nl(fun(B,A),fun(fun(num,B),fun(num,A)),H),F32),Num) ).

% num.case_distrib
tff(fact_5455_in__set__nthsD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I6: set(nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),nths(A,Xs,I6)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_nthsD
tff(fact_5456_notin__set__nthsI,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I6: set(nat)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ~ member(A,Xb,aa(list(A),set(A),set2(A),nths(A,Xs,I6))) ) ).

% notin_set_nthsI
tff(fact_5457_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_5458_distinct__nthsI,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( distinct(A,Xs)
     => distinct(A,nths(A,Xs,I6)) ) ).

% distinct_nthsI
tff(fact_5459_Diff__triv,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = A5 ) ) ).

% Diff_triv
tff(fact_5460_Int__Diff__disjoint,axiom,
    ! [A: $tType,A5: set(A),B5: 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)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_5461_Diff__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).

% Diff_eq
tff(fact_5462_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X2: num] : case_num(A,F1,F22,F32,aa(num,num,bit0,X2)) = aa(num,A,F22,X2) ).

% verit_eq_simplify(17)
tff(fact_5463_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A)] : case_num(A,F1,F22,F32,one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_5464_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I6))),aa(list(A),set(A),set2(A),Xs)) ).

% set_nths_subset
tff(fact_5465_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => member(nat,I2,I6) )
     => ( nths(A,Xs,I6) = Xs ) ) ).

% nths_all
tff(fact_5466_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_5467_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D3: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C3,D3)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),C3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),A2) ) ) ) ).

% Ioc_disjoint
tff(fact_5468_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_nm(fun(A,B),fun(set(A),fun(A,B)),G3),B5)),A5) ) ) ) ).

% prod.inter_restrict
tff(fact_5469_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T2: set(A),S3: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,finite_finite(A),S3)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,H,I2) = zero_zero(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T2))
                   => ( aa(A,B,G3,I2) = zero_zero(B) ) )
               => ( ! [X4: A] :
                      ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),T2))
                     => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
                 => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),S3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),T2) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_5470_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K2: A,Xb: A] :
          aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,K2)),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),K2),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_5471_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))) ) ) ) ).

% sum.Int_Diff
tff(fact_5472_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))) ) ) ) ).

% prod.Int_Diff
tff(fact_5473_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T2: set(A),S3: set(A),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),T2)
         => ( aa(set(A),$o,finite_finite(A),S3)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T2),S3))
                 => ( aa(A,B,H,I2) = one_one(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T2))
                   => ( aa(A,B,G3,I2) = one_one(B) ) )
               => ( ! [X4: A] :
                      ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),T2))
                     => ( aa(A,B,G3,X4) = aa(A,B,H,X4) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T2) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_5474_card__Diff__subset__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ).

% card_Diff_subset_Int
tff(fact_5475_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),P: fun(A,$o),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_nn(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G3)),A5) = 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)),A5),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% sum.If_cases
tff(fact_5476_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I6)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_no(list(A),fun(set(nat),fun(nat,$o)),Xs),I6))) ).

% length_nths
tff(fact_5477_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),P: fun(A,$o),H: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_np(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G3)),A5) = 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)),A5),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% prod.If_cases
tff(fact_5478_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A5: set(A),F3: fun(A,B),B2: B] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( divide_divide(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A5),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_nq(fun(A,B),fun(B,fun(A,B)),F3),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_nr(fun(A,B),fun(B,fun(A,$o)),F3),B2))))),divide_divide(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_ns(fun(A,B),fun(B,fun(A,$o)),F3),B2)))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_5479_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),aa(list(list(A)),list(list(A)),removeAll(list(A),nil(A)),Xs))
        & ! [Ys4: list(A)] :
            ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys4) )
        & ! [Ys4: list(A),Zs3: list(A)] :
            ( ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & 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_5480_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_5481_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_nd(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_5482_map__option__eq__Some,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xo: option(B),Y: A] :
      ( ( aa(option(B),option(A),map_option(B,A,F3),Xo) = aa(A,option(A),some(A),Y) )
    <=> ? [Z2: B] :
          ( ( Xo = aa(B,option(B),some(B),Z2) )
          & ( aa(B,A,F3,Z2) = Y ) ) ) ).

% map_option_eq_Some
tff(fact_5483_list__update__nonempty,axiom,
    ! [A: $tType,Xs: list(A),K2: nat,Xb: A] :
      ( ( list_update(A,Xs,K2,Xb) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% list_update_nonempty
tff(fact_5484_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xb: option(B)] :
      ( ( none(A) = aa(option(B),option(A),map_option(B,A,F3),Xb) )
    <=> ( Xb = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_5485_map__option__is__None,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Opt: option(B)] :
      ( ( aa(option(B),option(A),map_option(B,A,F3),Opt) = none(A) )
    <=> ( Opt = none(B) ) ) ).

% map_option_is_None
tff(fact_5486_option_Omap__disc__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A2: option(B)] :
      ( ( aa(option(B),option(A),map_option(B,A,F3),A2) = none(A) )
    <=> ( A2 = none(B) ) ) ).

% option.map_disc_iff
tff(fact_5487_concat__replicate__trivial,axiom,
    ! [A: $tType,I: nat] : concat(A,replicate(list(A),I,nil(A))) = nil(A) ).

% concat_replicate_trivial
tff(fact_5488_Nil__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( member(list(A),nil(A),shuffles(A,Xs,Ys))
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_in_shuffles
tff(fact_5489_nths__nil,axiom,
    ! [A: $tType,A5: set(nat)] : nths(A,nil(A),A5) = nil(A) ).

% nths_nil
tff(fact_5490_enumerate__simps_I1_J,axiom,
    ! [A: $tType,Nb: nat] : enumerate(A,Nb,nil(A)) = nil(product_prod(nat,A)) ).

% enumerate_simps(1)
tff(fact_5491_rotate1__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),rotate1(A),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rotate1_is_Nil_conv
tff(fact_5492_finite__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(set(list(A)),$o,finite_finite(list(A)),shuffles(A,Xs,Ys)) ).

% finite_shuffles
tff(fact_5493_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_5494_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_5495_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_5496_replicate__empty,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( replicate(A,Nb,Xb) = nil(A) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_5497_empty__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( nil(A) = replicate(A,Nb,Xb) )
    <=> ( Nb = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_5498_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(set(A),list(A),linord4507533701916653071of_set(A),bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_5499_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_5500_case__map__option,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: A,H: fun(B,A),F3: fun(C,B),Xb: option(C)] : case_option(A,B,G3,H,aa(option(C),option(B),map_option(C,B,F3),Xb)) = case_option(A,C,G3,aa(fun(C,B),fun(C,A),comp(B,A,C,H),F3),Xb) ).

% case_map_option
tff(fact_5501_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( concat(A,Xss) = nil(A) )
    <=> ! [X3: list(A)] :
          ( member(list(A),X3,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X3 = nil(A) ) ) ) ).

% concat_eq_Nil_conv
tff(fact_5502_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( nil(A) = concat(A,Xss) )
    <=> ! [X3: list(A)] :
          ( member(list(A),X3,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X3 = nil(A) ) ) ) ).

% Nil_eq_concat_conv
tff(fact_5503_nths__empty,axiom,
    ! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).

% nths_empty
tff(fact_5504_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_5505_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = nil(A) )
          <=> ( A5 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_5506_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B)),X: A,Xa: B] :
      ( aa(B,$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_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),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))),R),S3)) ) ).

% inf_Int_eq2
tff(fact_5507_map__option__cong,axiom,
    ! [B: $tType,A: $tType,Xb: option(A),Y: option(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( Xb = Y )
     => ( ! [A4: A] :
            ( ( Y = aa(A,option(A),some(A),A4) )
           => ( aa(A,B,F3,A4) = aa(A,B,G3,A4) ) )
       => ( aa(option(A),option(B),map_option(A,B,F3),Xb) = aa(option(A),option(B),map_option(A,B,G3),Y) ) ) ) ).

% map_option_cong
tff(fact_5508_option_Osimps_I9_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X2: B] : aa(option(B),option(A),map_option(B,A,F3),aa(B,option(B),some(B),X2)) = aa(A,option(A),some(A),aa(B,A,F3,X2)) ).

% option.simps(9)
tff(fact_5509_option_Osimps_I8_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] : aa(option(B),option(A),map_option(B,A,F3),none(B)) = none(A) ).

% option.simps(8)
tff(fact_5510_distinct_Osimps_I1_J,axiom,
    ! [A: $tType] : distinct(A,nil(A)) ).

% distinct.simps(1)
tff(fact_5511_insort__not__Nil,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),A2: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),A2),Xs) != nil(A) ) ).

% insort_not_Nil
tff(fact_5512_option_Omap__ident,axiom,
    ! [A: $tType,Ta: option(A)] : aa(option(A),option(A),map_option(A,A,aTP_Lamp_nt(A,A)),Ta) = Ta ).

% option.map_ident
tff(fact_5513_concat_Osimps_I1_J,axiom,
    ! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).

% concat.simps(1)
tff(fact_5514_list__update__code_I1_J,axiom,
    ! [A: $tType,I: nat,Y: A] : list_update(A,nil(A),I,Y) = nil(A) ).

% list_update_code(1)
tff(fact_5515_list__update_Osimps_I1_J,axiom,
    ! [A: $tType,I: nat,V: A] : list_update(A,nil(A),I,V) = nil(A) ).

% list_update.simps(1)
tff(fact_5516_product_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Uu2: list(B)] : product(A,B,nil(A),Uu2) = nil(product_prod(A,B)) ).

% product.simps(1)
tff(fact_5517_shuffles__commutes,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : shuffles(A,Xs,Ys) = shuffles(A,Ys,Xs) ).

% shuffles_commutes
tff(fact_5518_Nil__in__shufflesI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = nil(A) )
     => ( ( Ys = nil(A) )
       => member(list(A),nil(A),shuffles(A,Xs,Ys)) ) ) ).

% Nil_in_shufflesI
tff(fact_5519_shuffles_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),insert(list(A),Ys),bot_bot(set(list(A)))) ).

% shuffles.simps(1)
tff(fact_5520_shuffles_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] : shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),insert(list(A),Xs),bot_bot(set(list(A)))) ).

% shuffles.simps(2)
tff(fact_5521_remove1_Osimps_I1_J,axiom,
    ! [A: $tType,Xb: A] : remove1(A,Xb,nil(A)) = nil(A) ).

% remove1.simps(1)
tff(fact_5522_rotate1_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rotate1(A),nil(A)) = nil(A) ).

% rotate1.simps(1)
tff(fact_5523_removeAll_Osimps_I1_J,axiom,
    ! [A: $tType,Xb: A] : aa(list(A),list(A),removeAll(A,Xb),nil(A)) = nil(A) ).

% removeAll.simps(1)
tff(fact_5524_map__option_Ocompositionality,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G3: fun(C,B),Option: option(C)] : aa(option(B),option(A),map_option(B,A,F3),aa(option(C),option(B),map_option(C,B,G3),Option)) = aa(option(C),option(A),map_option(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F3),G3)),Option) ).

% map_option.compositionality
tff(fact_5525_option_Omap__comp,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: fun(B,A),F3: fun(C,B),V: option(C)] : aa(option(B),option(A),map_option(B,A,G3),aa(option(C),option(B),map_option(C,B,F3),V)) = aa(option(C),option(A),map_option(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,G3),F3)),V) ).

% option.map_comp
tff(fact_5526_map__option_Ocomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),G3: 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,F3)),map_option(A,C,G3)) = map_option(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,F3),G3)) ).

% map_option.comp
tff(fact_5527_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_5528_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_5529_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_5530_option_Omap__sel,axiom,
    ! [B: $tType,A: $tType,A2: option(A),F3: fun(A,B)] :
      ( ( A2 != none(A) )
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),map_option(A,B,F3),A2)) = aa(A,B,F3,aa(option(A),A,the2(A),A2)) ) ) ).

% option.map_sel
tff(fact_5531_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys))
     => ( aa(list(A),nat,size_size(list(A)),Zs) = 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_5532_replicate__0,axiom,
    ! [A: $tType,Xb: A] : replicate(A,zero_zero(nat),Xb) = nil(A) ).

% replicate_0
tff(fact_5533_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,nat)] : aa(list(A),nat,size_list(A,Xb),nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_5534_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_5535_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_5536_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_5537_option_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,nat),G3: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),comp(option(B),nat,option(A),size_option(B,F3)),map_option(A,B,G3)) = size_option(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F3),G3)) ).

% option.size_gen_o_map
tff(fact_5538_map__option__case,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Y: option(B)] : aa(option(B),option(A),map_option(B,A,F3),Y) = case_option(option(A),B,none(A),aTP_Lamp_nu(fun(B,A),fun(B,option(A)),F3),Y) ).

% map_option_case
tff(fact_5539_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: 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)) )
         => ( member(list(A),Zs,shuffles(A,Xs,Ys))
           => distinct(A,Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_5540_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,B),X: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),aTP_Lamp_nv(A,option(C))),X) = none(B) ).

% map_option_o_empty
tff(fact_5541_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_nd(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_5542_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_5543_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_5544_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_5545_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_5546_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_5547_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_5548_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_5549_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_5550_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_5551_and__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num,Q3: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb) = aa(num,option(num),some(num),Q3) )
        <=> ( 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),Q3) ) ) ) ).

% and_num_eq_Some_iff
tff(fact_5552_xor__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Mb: num,Nb: num,Q3: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,Mb),Nb) = aa(num,option(num),some(num),Q3) )
        <=> ( 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),Q3) ) ) ) ).

% xor_num_eq_Some_iff
tff(fact_5553_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_5554_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_5555_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_5556_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_5557_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_5558_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_5559_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_5560_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_5561_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_5562_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_5563_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_5564_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_nd(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(9)
tff(fact_5565_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_5566_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_5567_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_5568_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_5569_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),insert(list(A),nil(A)),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_5570_option_Orec__o__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,G3: B,Ga: fun(C,B),F3: fun(A,C)] : aa(fun(option(A),option(C)),fun(option(A),B),comp(option(C),B,option(A),rec_option(B,C,G3,Ga)),map_option(A,C,F3)) = rec_option(B,A,G3,aa(fun(A,C),fun(A,B),aTP_Lamp_nw(fun(C,B),fun(fun(A,C),fun(A,B)),Ga),F3)) ).

% option.rec_o_map
tff(fact_5571_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = aa(fun(real,$o),set(real),collect(real),aTP_Lamp_nx(real,$o)) ).

% Rats_eq_int_div_nat
tff(fact_5572_rat__floor__lemma,axiom,
    ! [A2: int,B2: int] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),divide_divide(int,A2,B2))),fract(A2,B2))
      & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A2,B2)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A2,B2)),one_one(int)))) ) ).

% rat_floor_lemma
tff(fact_5573_Rats__abs__iff,axiom,
    ! [Xb: real] :
      ( member(real,aa(real,real,abs_abs(real),Xb),field_char_0_Rats(real))
    <=> member(real,Xb,field_char_0_Rats(real)) ) ).

% Rats_abs_iff
tff(fact_5574_mult__rat,axiom,
    ! [A2: int,B2: int,C3: int,D3: int] : aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),fract(A2,B2)),fract(C3,D3)) = fract(aa(int,int,aa(int,fun(int,int),times_times(int),A2),C3),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ).

% mult_rat
tff(fact_5575_divide__rat,axiom,
    ! [A2: int,B2: int,C3: int,D3: int] : divide_divide(rat,fract(A2,B2),fract(C3,D3)) = fract(aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) ).

% divide_rat
tff(fact_5576_less__rat,axiom,
    ! [B2: int,D3: int,A2: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A2,B2)),fract(C3,D3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))) ) ) ) ).

% less_rat
tff(fact_5577_add__rat,axiom,
    ! [B2: int,D3: int,A2: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),fract(A2,B2)),fract(C3,D3)) = fract(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ) ) ) ).

% add_rat
tff(fact_5578_le__rat,axiom,
    ! [B2: int,D3: int,A2: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),fract(A2,B2)),fract(C3,D3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3))) ) ) ) ).

% le_rat
tff(fact_5579_diff__rat,axiom,
    ! [B2: int,D3: int,A2: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),fract(A2,B2)),fract(C3,D3)) = fract(aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ) ) ) ).

% diff_rat
tff(fact_5580_sgn__rat,axiom,
    ! [A2: int,B2: int] : aa(rat,rat,sgn_sgn(rat),fract(A2,B2)) = aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),A2)),aa(int,int,sgn_sgn(int),B2))) ).

% sgn_rat
tff(fact_5581_Rats__no__bot__less,axiom,
    ! [Xb: real] :
    ? [X4: real] :
      ( member(real,X4,field_char_0_Rats(real))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),Xb) ) ).

% Rats_no_bot_less
tff(fact_5582_Rats__dense__in__real,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => ? [X4: real] :
          ( member(real,X4,field_char_0_Rats(real))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),X4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),Y) ) ) ).

% Rats_dense_in_real
tff(fact_5583_Rats__power,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] :
          ( member(A,A2,field_char_0_Rats(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),field_char_0_Rats(A)) ) ) ).

% Rats_power
tff(fact_5584_Rats__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,field_char_0_Rats(A))
         => ( member(A,B2,field_char_0_Rats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_mult
tff(fact_5585_Rats__number__of,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [W: num] : member(A,aa(num,A,numeral_numeral(A),W),field_char_0_Rats(A)) ) ).

% Rats_number_of
tff(fact_5586_Rats__no__top__le,axiom,
    ! [Xb: real] :
    ? [X4: real] :
      ( member(real,X4,field_char_0_Rats(real))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),X4) ) ).

% Rats_no_top_le
tff(fact_5587_Rats__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,field_char_0_Rats(A))
         => ( member(A,B2,field_char_0_Rats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_add
tff(fact_5588_Rats__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => member(A,one_one(A),field_char_0_Rats(A)) ) ).

% Rats_1
tff(fact_5589_Rats__diff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,field_char_0_Rats(A))
         => ( member(A,B2,field_char_0_Rats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_diff
tff(fact_5590_Rats__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,field_char_0_Rats(A))
         => ( member(A,B2,field_char_0_Rats(A))
           => member(A,divide_divide(A,A2,B2),field_char_0_Rats(A)) ) ) ) ).

% Rats_divide
tff(fact_5591_eq__rat_I1_J,axiom,
    ! [B2: int,D3: int,A2: int,C3: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D3 != zero_zero(int) )
       => ( ( fract(A2,B2) = fract(C3,D3) )
        <=> ( aa(int,int,aa(int,fun(int,int),times_times(int),A2),D3) = aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2) ) ) ) ) ).

% eq_rat(1)
tff(fact_5592_mult__rat__cancel,axiom,
    ! [C3: int,A2: int,B2: int] :
      ( ( C3 != zero_zero(int) )
     => ( fract(aa(int,int,aa(int,fun(int,int),times_times(int),C3),A2),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B2)) = fract(A2,B2) ) ) ).

% mult_rat_cancel
tff(fact_5593_eq__rat_I2_J,axiom,
    ! [A2: int] : fract(A2,zero_zero(int)) = fract(zero_zero(int),one_one(int)) ).

% eq_rat(2)
tff(fact_5594_Fract__of__nat__eq,axiom,
    ! [K2: nat] : fract(aa(nat,int,semiring_1_of_nat(int),K2),one_one(int)) = aa(nat,rat,semiring_1_of_nat(rat),K2) ).

% Fract_of_nat_eq
tff(fact_5595_One__rat__def,axiom,
    one_one(rat) = fract(one_one(int),one_one(int)) ).

% One_rat_def
tff(fact_5596_Fract__of__int__eq,axiom,
    ! [K2: int] : fract(K2,one_one(int)) = aa(int,rat,ring_1_of_int(rat),K2) ).

% Fract_of_int_eq
tff(fact_5597_option_Osimps_I7_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),X2: B] : aa(option(B),A,rec_option(A,B,F1,F22),aa(B,option(B),some(B),X2)) = aa(B,A,F22,X2) ).

% option.simps(7)
tff(fact_5598_option_Osimps_I6_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,A)] : aa(option(B),A,rec_option(A,B,F1,F22),none(B)) = F1 ).

% option.simps(6)
tff(fact_5599_Zero__rat__def,axiom,
    zero_zero(rat) = fract(zero_zero(int),one_one(int)) ).

% Zero_rat_def
tff(fact_5600_rat__number__expand_I3_J,axiom,
    ! [K2: num] : aa(num,rat,numeral_numeral(rat),K2) = fract(aa(num,int,numeral_numeral(int),K2),one_one(int)) ).

% rat_number_expand(3)
tff(fact_5601_rat__number__collapse_I3_J,axiom,
    ! [W: num] : fract(aa(num,int,numeral_numeral(int),W),one_one(int)) = aa(num,rat,numeral_numeral(rat),W) ).

% rat_number_collapse(3)
tff(fact_5602_one__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),fract(A2,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),A2) ) ) ).

% one_less_Fract_iff
tff(fact_5603_Fract__less__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A2,B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),B2) ) ) ).

% Fract_less_one_iff
tff(fact_5604_Rats__eq__int__div__int,axiom,
    field_char_0_Rats(real) = aa(fun(real,$o),set(real),collect(real),aTP_Lamp_ny(real,$o)) ).

% Rats_eq_int_div_int
tff(fact_5605_rat__number__collapse_I5_J,axiom,
    fract(aa(int,int,uminus_uminus(int),one_one(int)),one_one(int)) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ).

% rat_number_collapse(5)
tff(fact_5606_Fract__add__one,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero(int) )
     => ( fract(aa(int,int,aa(int,fun(int,int),plus_plus(int),Mb),Nb),Nb) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),fract(Mb,Nb)),one_one(rat)) ) ) ).

% Fract_add_one
tff(fact_5607_one__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),fract(A2,B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2) ) ) ).

% one_le_Fract_iff
tff(fact_5608_Fract__le__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),fract(A2,B2)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),B2) ) ) ).

% Fract_le_one_iff
tff(fact_5609_rat__number__expand_I5_J,axiom,
    ! [K2: num] : aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K2)) = fract(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K2)),one_one(int)) ).

% rat_number_expand(5)
tff(fact_5610_rat__number__collapse_I4_J,axiom,
    ! [W: num] : fract(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W)),one_one(int)) = aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W)) ).

% rat_number_collapse(4)
tff(fact_5611_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C3: nat,Xb: nat,Y: nat] :
      image(nat,nat,aTP_Lamp_nz(nat,fun(nat,nat),C3),set_or7035219750837199246ssThan(nat,Xb,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C3),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xb),C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C3)),
        $ite(aa(nat,$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_5612_ring__1__class_Oof__int__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_of_int(A) = aa(fun(product_prod(nat,nat),A),fun(int,A),map_fun(int,product_prod(nat,nat),A,A,rep_Integ,id(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_mg(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_5613_sorted__key__list__of__set__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B)] : linord144544945434240204of_set(A,B,F3) = finite_folding_F(A,list(A),linorder_insort_key(A,B,F3),nil(A)) ) ).

% sorted_key_list_of_set_def
tff(fact_5614_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_5615_case__prod__Pair,axiom,
    ! [B: $tType,A: $tType] : 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)) = id(product_prod(A,B)) ).

% case_prod_Pair
tff(fact_5616_id__funpow,axiom,
    ! [A: $tType,Nb: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),id(A)) = id(A) ).

% id_funpow
tff(fact_5617_bij__betw__Suc,axiom,
    ! [M6: set(nat),N3: set(nat)] :
      ( bij_betw(nat,nat,suc,M6,N3)
    <=> ( image(nat,nat,suc,M6) = N3 ) ) ).

% bij_betw_Suc
tff(fact_5618_apfst__id,axiom,
    ! [B: $tType,A: $tType] : product_apfst(A,A,B,id(A)) = id(product_prod(A,B)) ).

% apfst_id
tff(fact_5619_apsnd__id,axiom,
    ! [B: $tType,A: $tType] : aa(fun(B,B),fun(product_prod(A,B),product_prod(A,B)),product_apsnd(B,B,A),id(B)) = id(product_prod(A,B)) ).

% apsnd_id
tff(fact_5620_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(A)] : image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),S3) = S3 ) ).

% image_add_0
tff(fact_5621_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I: A,J: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),K2),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2)) ) ).

% image_add_atLeastAtMost
tff(fact_5622_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A2: A,B2: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),D3),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),A2)) ) ).

% image_diff_atLeastAtMost
tff(fact_5623_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I: A,J: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),K2),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2)) ) ).

% image_add_atLeastLessThan
tff(fact_5624_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C3: A,A2: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),C3),set_ord_atMost(A,A2)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)) ) ).

% image_add_atMost
tff(fact_5625_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A5: set(A),B5: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A2),A5,B5)
        <=> ( image(A,A,aa(A,fun(A,A),plus_plus(A),A2),A5) = B5 ) ) ) ).

% bij_betw_add
tff(fact_5626_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C3: A,A2: A,B2: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),C3),set_or3652927894154168847AtMost(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ).

% image_add_greaterThanAtMost
tff(fact_5627_push__bit__0__id,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4730199178511100633sh_bit(A,zero_zero(nat)) = id(A) ) ) ).

% push_bit_0_id
tff(fact_5628_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_5629_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image(nat,nat,suc,set_or1337092689740270186AtMost(nat,I,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_5630_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image(nat,nat,suc,set_or7035219750837199246ssThan(nat,I,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_5631_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I: A,J: A] : image(A,A,aTP_Lamp_oa(A,fun(A,A),K2),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2)) ) ).

% image_add_atLeastAtMost'
tff(fact_5632_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D3: A,A2: A,B2: A] : image(A,A,aTP_Lamp_ob(A,fun(A,A),D3),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_5633_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I: A,J: A] : image(A,A,aTP_Lamp_oa(A,fun(A,A),K2),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K2),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K2)) ) ).

% image_add_atLeastLessThan'
tff(fact_5634_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A,B2: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),C3),set_or7035219750837199246ssThan(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)) ) ).

% image_diff_atLeastLessThan
tff(fact_5635_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A,B2: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),C3),set_or3652927894154168847AtMost(A,A2,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_5636_comp__the__Some,axiom,
    ! [A: $tType] : aa(fun(A,option(A)),fun(A,A),comp(option(A),A,A,the2(A)),some(A)) = id(A) ).

% comp_the_Some
tff(fact_5637_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( image(A,A,aa(A,fun(A,A),times_times(A),D3),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_5638_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D3)
         => ( image(A,A,aTP_Lamp_oc(A,fun(A,A),D3),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,divide_divide(A,A2,D3),divide_divide(A,B2,D3)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_5639_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] : aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_od(fun(B,A),fun(set(B),fun(A,$o)),F3),A5)) = image(B,A,F3,A5) ).

% Setcompr_eq_image
tff(fact_5640_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P: fun(B,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_oe(fun(B,A),fun(fun(B,$o),fun(A,$o)),F3),P)) = image(B,A,F3,aa(fun(B,$o),set(B),collect(B),P)) ).

% setcompr_eq_image
tff(fact_5641_map__option_Oidentity,axiom,
    ! [A: $tType] : map_option(A,A,aTP_Lamp_nt(A,A)) = id(option(A)) ).

% map_option.identity
tff(fact_5642_option_Omap__id0,axiom,
    ! [A: $tType] : map_option(A,A,id(A)) = id(option(A)) ).

% option.map_id0
tff(fact_5643_option_Omap__id,axiom,
    ! [A: $tType,Ta: option(A)] : aa(option(A),option(A),map_option(A,A,id(A)),Ta) = Ta ).

% option.map_id
tff(fact_5644_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S2: set(A),Ta: 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)),S2),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),S2)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),Ta)) ) ).

% translation_Int
tff(fact_5645_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),B5: set(B)] : aa(set(A),$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)),image(B,A,F3,A5)),image(B,A,F3,B5))),image(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B5))) ).

% image_diff_subset
tff(fact_5646_zero__notin__Suc__image,axiom,
    ! [A5: set(nat)] : ~ member(nat,zero_zero(nat),image(nat,nat,suc,A5)) ).

% zero_notin_Suc_image
tff(fact_5647_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F3: fun(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)),F3) = id(A) ).

% funpow_simps_right(1)
tff(fact_5648_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S2: set(A),Ta: 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)),S2),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),S2)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),Ta)) ) ).

% translation_diff
tff(fact_5649_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: 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)),image(A,A,aa(A,fun(A,A),plus_plus(A),A2),Ta)) ) ).

% translation_Compl
tff(fact_5650_less__int__def,axiom,
    ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$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_mi(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int_def
tff(fact_5651_less__eq__int__def,axiom,
    ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(int,fun(int,$o)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(int,$o),rep_Integ,map_fun(int,product_prod(nat,nat),$o,$o,rep_Integ,id($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$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_mk(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int_def
tff(fact_5652_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
    <=> ? [N4: nat,F5: fun(nat,A)] : A5 = image(nat,A,F5,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),N4))) ) ).

% finite_conv_nat_seg_image
tff(fact_5653_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A5: set(A),F3: fun(nat,A),Nb: nat] :
      ( ( A5 = image(nat,A,F3,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Nb))) )
     => aa(set(A),$o,finite_finite(A),A5) ) ).

% nat_seg_image_imp_finite
tff(fact_5654_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S2: set(A),Ta: set(A)] : image(A,A,aTP_Lamp_of(A,fun(A,A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),image(A,A,aTP_Lamp_of(A,fun(A,A),A2),S2)),image(A,A,aTP_Lamp_of(A,fun(A,A),A2),Ta)) ) ).

% translation_subtract_Int
tff(fact_5655_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S2: set(A),Ta: set(A)] : image(A,A,aTP_Lamp_of(A,fun(A,A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),Ta)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),image(A,A,aTP_Lamp_of(A,fun(A,A),A2),S2)),image(A,A,aTP_Lamp_of(A,fun(A,A),A2),Ta)) ) ).

% translation_subtract_diff
tff(fact_5656_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: set(A)] : image(A,A,aTP_Lamp_of(A,fun(A,A),A2),aa(set(A),set(A),uminus_uminus(set(A)),Ta)) = aa(set(A),set(A),uminus_uminus(set(A)),image(A,A,aTP_Lamp_of(A,fun(A,A),A2),Ta)) ) ).

% translation_subtract_Compl
tff(fact_5657_nat__def,axiom,
    nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))) ).

% nat_def
tff(fact_5658_card__image__le,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),image(A,B,F3,A5))),aa(set(A),nat,finite_card(A),A5)) ) ).

% card_image_le
tff(fact_5659_fst__diag__id,axiom,
    ! [A: $tType,Z: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_fst(A,A)),aTP_Lamp_lx(A,product_prod(A,A))),Z) = aa(A,A,id(A),Z) ).

% fst_diag_id
tff(fact_5660_snd__diag__id,axiom,
    ! [A: $tType,Z: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),comp(product_prod(A,A),A,A,product_snd(A,A)),aTP_Lamp_lx(A,product_prod(A,A))),Z) = aa(A,A,id(A),Z) ).

% snd_diag_id
tff(fact_5661_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A5: set(A),H: fun(A,B),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ! [X4: A,Y3: A] :
                ( member(A,X4,A5)
               => ( member(A,Y3,A5)
                 => ( ( X4 != Y3 )
                   => ( ( aa(A,B,H,X4) = aa(A,B,H,Y3) )
                     => ( aa(B,C,G3,aa(A,B,H,X4)) = one_one(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G3),image(A,B,H,A5)) = 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,G3),H)),A5) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_5662_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B5),image(A,B,F3,A5))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B5)),aa(set(A),nat,finite_card(A),A5)) ) ) ).

% surj_card_le
tff(fact_5663_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image(nat,nat,suc,set_ord_lessThan(nat,Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),Nb) ).

% image_Suc_lessThan
tff(fact_5664_image__Suc__atMost,axiom,
    ! [Nb: nat] : image(nat,nat,suc,set_ord_atMost(nat,Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,Nb)) ).

% image_Suc_atMost
tff(fact_5665_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)),image(nat,nat,suc,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_5666_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)),image(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_5667_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image(nat,nat,suc,set_ord_lessThan(nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_5668_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_ord_atMost(nat,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image(nat,nat,suc,set_ord_atMost(nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_5669_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I6: set(A),G3: fun(C,B),F3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite(A),I6)
         => ( ! [I2: A] :
                ( member(A,I2,I6)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G3,aa(A,C,F3,I2))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,groups7311177749621191930dd_sum(C,B,G3),image(A,C,F3,I6))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,G3),F3)),I6)) ) ) ) ).

% sum_image_le
tff(fact_5670_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,Xb: A,Y: A] :
          image(A,A,aa(A,fun(A,A),times_times(A),C3),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),C3),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),C3),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Xb)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_5671_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,Xb: A,Y: A] :
          image(A,A,aTP_Lamp_og(A,fun(A,A),C3),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)),C3),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C3))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_5672_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C3: A,A2: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),Mb),C3),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)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C3)),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)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C3))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_5673_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C3: A,A2: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),Mb),C3),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)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),B2)),C3)),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)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Mb),A2)),C3))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_5674_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C3: A,A2: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_oj(A,fun(A,fun(A,A)),Mb),C3),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),divide_divide(A,A2,Mb)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,Mb)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,Mb)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Mb)),C3))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_5675_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Mb: A,C3: A,A2: A,B2: A] :
          image(A,A,aa(A,fun(A,A),aTP_Lamp_ok(A,fun(A,fun(A,A)),Mb),C3),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),divide_divide(A,A2,Mb)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,Mb)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,B2,Mb)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,A2,Mb)),C3))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_5676_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S3: set(A),R: set(B),G3: fun(A,B),F3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( aa(set(B),$o,finite_finite(B),R)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,G3,S3)),R)
             => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ol(fun(A,B),fun(fun(B,C),fun(A,C)),G3),F3)),S3) = 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_on(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S3),G3),F3)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_5677_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F3: fun(A,fun(B,B)),A5: set(A),Xb: A,Z: B] :
      ( finite_folding_on(A,B,S3,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S3)
       => ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),A5) = aa(B,B,aa(A,fun(B,B),F3,Xb),aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_5678_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite_folding_on(A,B,S3,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xb),A5)),S3)
       => ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),insert(A,Xb),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xb),aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_5679_positive__rat,axiom,
    ! [A2: int,B2: int] :
      ( aa(rat,$o,positive,fract(A2,B2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ).

% positive_rat
tff(fact_5680_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A5: set(product_prod(A,B)),F3: fun(A,fun(B,C))] :
      ( 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),A5)
     => member(C,aa(B,C,aa(A,fun(B,C),F3,A2),B2),image(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),A5)) ) ).

% pair_imageI
tff(fact_5681_None__notin__image__Some,axiom,
    ! [A: $tType,A5: set(A)] : ~ member(option(A),none(A),image(A,option(A),some(A),A5)) ).

% None_notin_image_Some
tff(fact_5682_Rat_Opositive__mult,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa(rat,$o,positive,Xb)
     => ( aa(rat,$o,positive,Y)
       => aa(rat,$o,positive,aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Xb),Y)) ) ) ).

% Rat.positive_mult
tff(fact_5683_in__image__insert__iff,axiom,
    ! [A: $tType,B5: set(set(A)),Xb: A,A5: set(A)] :
      ( ! [C7: set(A)] :
          ( member(set(A),C7,B5)
         => ~ member(A,Xb,C7) )
     => ( member(set(A),A5,image(set(A),set(A),insert(A,Xb),B5))
      <=> ( member(A,Xb,A5)
          & member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_5684_less__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xb),Y)
    <=> aa(rat,$o,positive,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Y),Xb)) ) ).

% less_rat_def
tff(fact_5685_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X6: set(A),A5: set(product_prod(A,B)),Y6: set(B),P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ( X6 = image(product_prod(A,B),A,product_fst(A,B),A5) )
     => ( ( Y6 = image(product_prod(A,B),B,product_snd(A,B),A5) )
       => ( ! [X4: A] :
              ( member(A,X4,X6)
             => ! [Xa3: B] :
                  ( member(B,Xa3,Y6)
                 => ( aa(B,$o,aa(A,fun(B,$o),P,X4),Xa3)
                   => aa(B,$o,aa(A,fun(B,$o),Q,X4),Xa3) ) ) )
         => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),$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,$o),P)))
           => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),$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),Q))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_5686_subset__subseqs,axiom,
    ! [A: $tType,X6: set(A),Xs: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),aa(list(A),set(A),set2(A),Xs))
     => member(set(A),X6,image(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ) ).

% subset_subseqs
tff(fact_5687_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image(int,int,aTP_Lamp_oo(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_5688_card__def,axiom,
    ! [A: $tType] : finite_card(A) = finite_folding_F(A,nat,aTP_Lamp_op(A,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_5689_Rat_Opositive_Orep__eq,axiom,
    ! [Xb: rat] :
      ( aa(rat,$o,positive,Xb)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb)))) ) ).

% Rat.positive.rep_eq
tff(fact_5690_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)
             => ? [C2: 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),C2),B2)
                  & ! [X: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),C2) )
                     => aa(A,$o,P,X) )
                  & ! [D6: A] :
                      ( ! [X4: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),D6) )
                         => aa(A,$o,P,X4) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D6),C2) ) ) ) ) ) ) ).

% complete_interval
tff(fact_5691_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs))
     => ( image(nat,A,nth(A,Xs),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),take(A,L,Xs)) ) ) ).

% nth_image
tff(fact_5692_take0,axiom,
    ! [A: $tType,X: list(A)] : take(A,zero_zero(nat),X) = nil(A) ).

% take0
tff(fact_5693_take__eq__Nil,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( take(A,Nb,Xs) = nil(A) )
    <=> ( ( Nb = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_5694_take__eq__Nil2,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,Nb,Xs) )
    <=> ( ( Nb = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_5695_take__all,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb)
     => ( take(A,Nb,Xs) = Xs ) ) ).

% take_all
tff(fact_5696_take__all__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( take(A,Nb,Xs) = Xs )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% take_all_iff
tff(fact_5697_nth__take,axiom,
    ! [A: $tType,I: nat,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Nb)
     => ( aa(nat,A,nth(A,take(A,Nb,Xs)),I) = aa(nat,A,nth(A,Xs),I) ) ) ).

% nth_take
tff(fact_5698_take__update__cancel,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => ( take(A,Nb,list_update(A,Xs,Mb,Y)) = take(A,Nb,Xs) ) ) ).

% take_update_cancel
tff(fact_5699_nths__upt__eq__take,axiom,
    ! [A: $tType,L: list(A),Nb: nat] : nths(A,L,set_ord_lessThan(nat,Nb)) = take(A,Nb,L) ).

% nths_upt_eq_take
tff(fact_5700_distinct__take,axiom,
    ! [A: $tType,Xs: list(A),I: nat] :
      ( distinct(A,Xs)
     => distinct(A,take(A,I,Xs)) ) ).

% distinct_take
tff(fact_5701_in__set__takeD,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),take(A,Nb,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_takeD
tff(fact_5702_take__update__swap,axiom,
    ! [A: $tType,Mb: nat,Xs: list(A),Nb: nat,Xb: A] : take(A,Mb,list_update(A,Xs,Nb,Xb)) = list_update(A,take(A,Mb,Xs),Nb,Xb) ).

% take_update_swap
tff(fact_5703_take__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ! [I2: nat] : take(A,I2,Xs) = take(A,I2,Ys)
     => ( Xs = Ys ) ) ).

% take_equalityI
tff(fact_5704_set__take__subset,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,Nb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_take_subset
tff(fact_5705_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_5706_take__Nil,axiom,
    ! [A: $tType,Nb: nat] : take(A,Nb,nil(A)) = nil(A) ).

% take_Nil
tff(fact_5707_set__take__subset__set__take,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,Mb,Xs))),aa(list(A),set(A),set2(A),take(A,Nb,Xs))) ) ).

% set_take_subset_set_take
tff(fact_5708_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A2: A] :
        ? [B4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B4)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),A2) ) ) ).

% ex_gt_or_lt
tff(fact_5709_nth__take__lemma,axiom,
    ! [A: $tType,K2: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),aa(list(A),nat,size_size(list(A)),Ys))
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K2)
             => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
         => ( take(A,K2,Xs) = take(A,K2,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_5710_take__bit__horner__sum__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,Bs: list($o)] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),Bs)) = groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),take($o,Nb,Bs)) ) ).

% take_bit_horner_sum_bit_eq
tff(fact_5711_Rat_Opositive__def,axiom,
    positive = aa(fun(product_prod(int,int),$o),fun(rat,$o),map_fun(rat,product_prod(int,int),$o,$o,rep_Rat,id($o)),aTP_Lamp_oq(product_prod(int,int),$o)) ).

% Rat.positive_def
tff(fact_5712_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2))
     => ~ ! [I2: nat] :
            ( aa(nat,$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),I2),aa(list(A),nat,size_size(list(A)),Ys))
             => ( ( take(A,I2,Xs) = take(A,I2,Ys) )
               => ~ 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),I2)),aa(nat,A,nth(A,Ys),I2)),R2) ) ) ) ) ).

% lex_take_index
tff(fact_5713_Some__image__these__eq,axiom,
    ! [A: $tType,A5: set(option(A))] : image(A,option(A),some(A),these(A,A5)) = aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_or(set(option(A)),fun(option(A),$o),A5)) ).

% Some_image_these_eq
tff(fact_5714_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_5715_these__insert__None,axiom,
    ! [A: $tType,A5: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),none(A)),A5)) = these(A,A5) ).

% these_insert_None
tff(fact_5716_these__insert__Some,axiom,
    ! [A: $tType,Xb: A,A5: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert(option(A),aa(A,option(A),some(A),Xb)),A5)) = aa(set(A),set(A),insert(A,Xb),these(A,A5)) ).

% these_insert_Some
tff(fact_5717_these__image__Some__eq,axiom,
    ! [A: $tType,A5: set(A)] : these(A,image(A,option(A),some(A),A5)) = A5 ).

% these_image_Some_eq
tff(fact_5718_in__these__eq,axiom,
    ! [A: $tType,Xb: A,A5: set(option(A))] :
      ( member(A,Xb,these(A,A5))
    <=> member(option(A),aa(A,option(A),some(A),Xb),A5) ) ).

% in_these_eq
tff(fact_5719_Nil2__notin__lex,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ 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),nil(A)),lex(A,R2)) ).

% Nil2_notin_lex
tff(fact_5720_Nil__notin__lex,axiom,
    ! [A: $tType,Ys: list(A),R2: set(product_prod(A,A))] : ~ 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)),nil(A)),Ys),lex(A,R2)) ).

% Nil_notin_lex
tff(fact_5721_these__empty__eq,axiom,
    ! [A: $tType,B5: set(option(A))] :
      ( ( these(A,B5) = bot_bot(set(A)) )
    <=> ( ( B5 = bot_bot(set(option(A))) )
        | ( B5 = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_5722_these__not__empty__eq,axiom,
    ! [A: $tType,B5: set(option(A))] :
      ( ( these(A,B5) != bot_bot(set(A)) )
    <=> ( ( B5 != bot_bot(set(option(A))) )
        & ( B5 != aa(set(option(A)),set(option(A)),insert(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_5723_Option_Othese__def,axiom,
    ! [A: $tType,A5: set(option(A))] : these(A,A5) = image(option(A),A,the2(A),aa(fun(option(A),$o),set(option(A)),collect(option(A)),aTP_Lamp_or(set(option(A)),fun(option(A),$o),A5))) ).

% Option.these_def
tff(fact_5724_of__rat__def,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_of_rat(A) = aa(fun(product_prod(int,int),A),fun(rat,A),map_fun(rat,product_prod(int,int),A,A,rep_Rat,id(A)),aTP_Lamp_os(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_5725_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
    <=> ? [Y5: A,N4: nat] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N4)),Y5),R2)
          & aa(nat,$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,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_5726_lenlex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = aa(fun(product_prod(list(A),list(A)),$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),$o),aTP_Lamp_ot(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lenlex_conv
tff(fact_5727_of__rat__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),one_one(rat)) = one_one(A) ) ) ).

% of_rat_1
tff(fact_5728_of__rat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( aa(rat,A,field_char_0_of_rat(A),A2) = one_one(A) )
        <=> ( A2 = one_one(rat) ) ) ) ).

% of_rat_eq_1_iff
tff(fact_5729_one__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat] :
          ( ( one_one(A) = aa(rat,A,field_char_0_of_rat(A),A2) )
        <=> ( one_one(rat) = A2 ) ) ) ).

% one_eq_of_rat_iff
tff(fact_5730_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] :
      ( 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)),nil(A)),Ns),lenlex(A,R2))
    <=> ( Ns != nil(A) ) ) ).

% Nil_lenlex_iff1
tff(fact_5731_of__rat__numeral__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [W: num] : aa(rat,A,field_char_0_of_rat(A),aa(num,rat,numeral_numeral(rat),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_rat_numeral_eq
tff(fact_5732_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R2)),zero_zero(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),zero_zero(rat)) ) ) ).

% of_rat_less_0_iff
tff(fact_5733_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2) ) ) ).

% zero_less_of_rat_iff
tff(fact_5734_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),R2) ) ) ).

% one_less_of_rat_iff
tff(fact_5735_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R2)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),one_one(rat)) ) ) ).

% of_rat_less_1_iff
tff(fact_5736_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),zero_zero(rat)),R2) ) ) ).

% zero_le_of_rat_iff
tff(fact_5737_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R2)),zero_zero(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),zero_zero(rat)) ) ) ).

% of_rat_le_0_iff
tff(fact_5738_of__rat__neg__one,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% of_rat_neg_one
tff(fact_5739_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),R2) ) ) ).

% one_le_of_rat_iff
tff(fact_5740_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R2)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),one_one(rat)) ) ) ).

% of_rat_le_1_iff
tff(fact_5741_of__rat__neg__numeral__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [W: num] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) ) ).

% of_rat_neg_numeral_eq
tff(fact_5742_listrel1__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S2)
     => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel1(A,S2)) ) ).

% listrel1_mono
tff(fact_5743_of__rat__power,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,Nb: nat] : aa(rat,A,field_char_0_of_rat(A),aa(nat,rat,aa(rat,fun(nat,rat),power_power(rat),A2),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(rat,A,field_char_0_of_rat(A),A2)),Nb) ) ).

% of_rat_power
tff(fact_5744_of__rat__dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),Y)
     => ? [Q2: rat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),aa(rat,real,field_char_0_of_rat(real),Q2))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(rat,real,field_char_0_of_rat(real),Q2)),Y) ) ) ).

% of_rat_dense
tff(fact_5745_of__rat__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat,S2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R2)),aa(rat,A,field_char_0_of_rat(A),S2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),S2) ) ) ).

% of_rat_less
tff(fact_5746_of__rat__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(rat,A,field_char_0_of_rat(A),A2)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_add
tff(fact_5747_of__rat__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat,S2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R2)),aa(rat,A,field_char_0_of_rat(A),S2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),S2) ) ) ).

% of_rat_less_eq
tff(fact_5748_of__rat__mult,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(rat,A,field_char_0_of_rat(A),A2)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_mult
tff(fact_5749_of__rat__diff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(rat,A,field_char_0_of_rat(A),A2)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_diff
tff(fact_5750_not__Nil__listrel1,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ 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)),nil(A)),Xs),listrel1(A,R2)) ).

% not_Nil_listrel1
tff(fact_5751_not__listrel1__Nil,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : ~ 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),nil(A)),listrel1(A,R2)) ).

% not_listrel1_Nil
tff(fact_5752_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( 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,R2))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_5753_of__rat__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),divide_divide(rat,A2,B2)) = divide_divide(A,aa(rat,A,field_char_0_of_rat(A),A2),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_divide
tff(fact_5754_lenlex__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R2)
     => ~ 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,R2)) ) ).

% lenlex_irreflexive
tff(fact_5755_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list(A),R2: set(product_prod(A,A))] : ~ 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)),Ns),nil(A)),lenlex(A,R2)) ).

% Nil_lenlex_iff2
tff(fact_5756_nonzero__of__rat__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [B2: rat,A2: rat] :
          ( ( B2 != zero_zero(rat) )
         => ( aa(rat,A,field_char_0_of_rat(A),divide_divide(rat,A2,B2)) = divide_divide(A,aa(rat,A,field_char_0_of_rat(A),A2),aa(rat,A,field_char_0_of_rat(A),B2)) ) ) ) ).

% nonzero_of_rat_divide
tff(fact_5757_of__rat__rat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [B2: int,A2: int] :
          ( ( B2 != zero_zero(int) )
         => ( aa(rat,A,field_char_0_of_rat(A),fract(A2,B2)) = divide_divide(A,aa(int,A,ring_1_of_int(A),A2),aa(int,A,ring_1_of_int(A),B2)) ) ) ) ).

% of_rat_rat
tff(fact_5758_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2))
     => aa(nat,$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_5759_of__rat_Orep__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xb: rat] : aa(rat,A,field_char_0_of_rat(A),Xb) = divide_divide(A,aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb))),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,Xb)))) ) ).

% of_rat.rep_eq
tff(fact_5760_listrel1p__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( listrel1p(A,R2,Xs,Ys)
    <=> 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),R2)))) ) ).

% listrel1p_def
tff(fact_5761_plus__rat__def,axiom,
    plus_plus(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_ou(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_5762_inverse__rat__def,axiom,
    inverse_inverse(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_ov(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_5763_one__rat__def,axiom,
    one_one(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).

% one_rat_def
tff(fact_5764_Fract_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      fract(Xaa,Xb) = aa(product_prod(int,int),rat,abs_Rat,
        $ite(Xb = 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),Xaa),Xb))) ).

% Fract.abs_eq
tff(fact_5765_zero__rat__def,axiom,
    zero_zero(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).

% zero_rat_def
tff(fact_5766_times__rat__def,axiom,
    times_times(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_ow(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_5767_plus__rat_Oabs__eq,axiom,
    ! [Xaa: product_prod(int,int),Xb: product_prod(int,int)] :
      ( ratrel(Xaa,Xaa)
     => ( ratrel(Xb,Xb)
       => ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xaa)),aa(product_prod(int,int),rat,abs_Rat,Xb)) = aa(product_prod(int,int),rat,abs_Rat,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(product_prod(int,int),int,product_fst(int,int),Xaa)),aa(product_prod(int,int),int,product_snd(int,int),Xb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xb)),aa(product_prod(int,int),int,product_snd(int,int),Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xaa)),aa(product_prod(int,int),int,product_snd(int,int),Xb)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_5768_inverse__rat_Oabs__eq,axiom,
    ! [Xb: product_prod(int,int)] :
      ( ratrel(Xb,Xb)
     => ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,Xb)) = aa(product_prod(int,int),rat,abs_Rat,
            $ite(aa(product_prod(int,int),int,product_fst(int,int),Xb) = 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(product_prod(int,int),int,product_snd(int,int),Xb)),aa(product_prod(int,int),int,product_fst(int,int),Xb)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_5769_Rat_Opositive_Oabs__eq,axiom,
    ! [Xb: product_prod(int,int)] :
      ( ratrel(Xb,Xb)
     => ( aa(rat,$o,positive,aa(product_prod(int,int),rat,abs_Rat,Xb))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xb)),aa(product_prod(int,int),int,product_snd(int,int),Xb))) ) ) ).

% Rat.positive.abs_eq
tff(fact_5770_ratrel__iff,axiom,
    ! [Xb: product_prod(int,int),Y: product_prod(int,int)] :
      ( ratrel(Xb,Y)
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),Xb) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Y) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xb)),aa(product_prod(int,int),int,product_snd(int,int),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Y)),aa(product_prod(int,int),int,product_snd(int,int),Xb)) ) ) ) ).

% ratrel_iff
tff(fact_5771_one__rat_Orsp,axiom,
    ratrel(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).

% one_rat.rsp
tff(fact_5772_zero__rat_Orsp,axiom,
    ratrel(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),zero_zero(int)),one_one(int))) ).

% zero_rat.rsp
tff(fact_5773_ratrel__def,axiom,
    ! [X: product_prod(int,int),Xa: product_prod(int,int)] :
      ( ratrel(X,Xa)
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X) != zero_zero(int) )
        & ( aa(product_prod(int,int),int,product_snd(int,int),Xa) != zero_zero(int) )
        & ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)) ) ) ) ).

% ratrel_def
tff(fact_5774_of__rat_Oabs__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xb: product_prod(int,int)] :
          ( ratrel(Xb,Xb)
         => ( aa(rat,A,field_char_0_of_rat(A),aa(product_prod(int,int),rat,abs_Rat,Xb)) = divide_divide(A,aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),Xb)),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),Xb))) ) ) ) ).

% of_rat.abs_eq
tff(fact_5775_times__rat_Oabs__eq,axiom,
    ! [Xaa: product_prod(int,int),Xb: product_prod(int,int)] :
      ( ratrel(Xaa,Xaa)
     => ( ratrel(Xb,Xb)
       => ( aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xaa)),aa(product_prod(int,int),rat,abs_Rat,Xb)) = aa(product_prod(int,int),rat,abs_Rat,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(product_prod(int,int),int,product_fst(int,int),Xaa)),aa(product_prod(int,int),int,product_fst(int,int),Xb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xaa)),aa(product_prod(int,int),int,product_snd(int,int),Xb)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_5776_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_ox(A,fun(A,$o))) = ord_max(A) ) ) ).

% dual_min
tff(fact_5777_card__Min__le__sum,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),image(A,nat,F3,A5)))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5)) ) ).

% card_Min_le_sum
tff(fact_5778_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat,Mb: nat] : modulo_modulo(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)),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),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,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_5779_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))
        <=> ( aa(A,$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),C3) ) ) ) ).

% min.bounded_iff
tff(fact_5780_min_Oabsorb2,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_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_5781_min_Oabsorb1,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_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_5782_min_Oabsorb3,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_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb3
tff(fact_5783_min_Oabsorb4,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_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_5784_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_5785_min__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Mb)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)) ).

% min_Suc_Suc
tff(fact_5786_min__0R,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_5787_min__0L,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),Nb) = zero_zero(nat) ).

% min_0L
tff(fact_5788_take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_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),ord_min(nat),Mb),Nb)),A2) ) ).

% take_bit_take_bit
tff(fact_5789_take__take,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] : take(A,Nb,take(A,Mb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),Mb),Xs) ).

% take_take
tff(fact_5790_signed__take__bit__signed__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_ri4674362597316999326ke_bit(A,Nb),A2)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)),A2) ) ).

% signed_take_bit_signed_take_bit
tff(fact_5791_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(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),U),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(1)
tff(fact_5792_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xb)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_5793_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xb)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_5794_min__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),zero_zero(A)) = zero_zero(A) ) ) ).

% min_0_1(2)
tff(fact_5795_min__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),one_one(A)) = zero_zero(A) ) ) ).

% min_0_1(1)
tff(fact_5796_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_5797_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_5798_length__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),take(A,Nb,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ).

% length_take
tff(fact_5799_take__bit__of__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Mb: nat,Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Mb),bit_se2239418461657761734s_mask(A,Nb)) = bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)) ) ).

% take_bit_of_mask
tff(fact_5800_take__replicate,axiom,
    ! [A: $tType,I: nat,K2: nat,Xb: A] : take(A,I,replicate(A,K2,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),K2),Xb) ).

% take_replicate
tff(fact_5801_min__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_min(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(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(2)
tff(fact_5802_min__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_min(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(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(3)
tff(fact_5803_min__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_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$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),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(4)
tff(fact_5804_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5))
            <=> ! [X3: A] :
                  ( member(A,X3,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X3) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_5805_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5))
            <=> ! [X3: A] :
                  ( member(A,X3,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X3) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_5806_min__Suc__numeral,axiom,
    ! [Nb: nat,K2: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K2)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),pred_numeral(K2))) ).

% min_Suc_numeral
tff(fact_5807_min__numeral__Suc,axiom,
    ! [K2: num,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K2)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K2)),Nb)) ).

% min_numeral_Suc
tff(fact_5808_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Min.remove
tff(fact_5809_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),insert(A,Xb),A5)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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_min(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ).

% Min.insert_remove
tff(fact_5810_of__nat__min,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_min(nat),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_5811_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_5812_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% min.strict_boundedE
tff(fact_5813_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_5814_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.strict_coboundedI1
tff(fact_5815_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.strict_coboundedI2
tff(fact_5816_nat__mult__min__right,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Nb),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(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),Q3)) ).

% nat_mult_min_right
tff(fact_5817_nat__mult__min__left,axiom,
    ! [Mb: nat,Nb: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Mb),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q3)) ).

% nat_mult_min_left
tff(fact_5818_min__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_min(A),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(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)) ) ).

% min_add_distrib_left
tff(fact_5819_min__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_min(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(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)) ) ).

% min_add_distrib_right
tff(fact_5820_min__diff,axiom,
    ! [Mb: nat,I: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),I)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),I)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb)),I) ).

% min_diff
tff(fact_5821_min__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_min(A),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(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)) ) ).

% min_diff_distrib_left
tff(fact_5822_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xb: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_5823_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xb: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_5824_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_5825_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xa: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),X),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa),X,Xa) ) ).

% min_def_raw
tff(fact_5826_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),A2,B2) ) ).

% min_def
tff(fact_5827_min__absorb1,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_min(A),Xb),Y) = Xb ) ) ) ).

% min_absorb1
tff(fact_5828_min__absorb2,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_min(A),Xb),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_5829_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C3),D3)) ) ) ) ).

% min.mono
tff(fact_5830_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.orderE
tff(fact_5831_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% min.orderI
tff(fact_5832_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))
         => ~ ( aa(A,$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),C3) ) ) ) ).

% min.boundedE
tff(fact_5833_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)) ) ) ) ).

% min.boundedI
tff(fact_5834_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.order_iff
tff(fact_5835_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2) ) ).

% min.cobounded1
tff(fact_5836_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2) ) ).

% min.cobounded2
tff(fact_5837_min_Oabsorb__iff1,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_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb_iff1
tff(fact_5838_min_Oabsorb__iff2,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_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_5839_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.coboundedI1
tff(fact_5840_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.coboundedI2
tff(fact_5841_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),Z)
        <=> ( aa(A,$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) ) ) ) ).

% min_le_iff_disj
tff(fact_5842_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),Xb) ) ) ) ).

% Min_le
tff(fact_5843_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ! [Y3: A] :
                ( member(A,Y3,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y3) )
           => ( member(A,Xb,A5)
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = Xb ) ) ) ) ) ).

% Min_eqI
tff(fact_5844_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),A2) ) ) ) ).

% Min.coboundedI
tff(fact_5845_concat__bit__assoc__sym,axiom,
    ! [Mb: nat,Nb: nat,K2: int,L: int,R2: int] : aa(int,int,bit_concat_bit(Mb,aa(int,int,bit_concat_bit(Nb,K2),L)),R2) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb),K2),aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_5846_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => 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),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).

% Min.boundedI
tff(fact_5847_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798350308766er_Min(A),A5))
             => ! [A8: A] :
                  ( member(A,A8,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A8) ) ) ) ) ) ).

% Min.boundedE
tff(fact_5848_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( Mb = aa(set(A),A,lattic643756798350308766er_Min(A),A5) )
            <=> ( member(A,Mb,A5)
                & ! [X3: A] :
                    ( member(A,X3,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X3) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_5849_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),Xb)
            <=> ? [X3: A] :
                  ( member(A,X3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xb) ) ) ) ) ) ).

% Min_le_iff
tff(fact_5850_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = Mb )
            <=> ( member(A,Mb,A5)
                & ! [X3: A] :
                    ( member(A,X3,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Mb),X3) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_5851_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),Xb)
            <=> ? [X3: A] :
                  ( member(A,X3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xb) ) ) ) ) ) ).

% Min_less_iff
tff(fact_5852_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ! [B4: A] :
                ( member(A,B4,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B4) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),insert(A,A2),A5)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_5853_take__bit__concat__bit__eq,axiom,
    ! [Mb: nat,Nb: nat,K2: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Mb),aa(int,int,bit_concat_bit(Nb,K2),L)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),Nb),K2),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb)),L)) ).

% take_bit_concat_bit_eq
tff(fact_5854_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Min.infinite
tff(fact_5855_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)),P2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).

% min_mult_distrib_right
tff(fact_5856_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)),P2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).

% max_mult_distrib_right
tff(fact_5857_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,Xb: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).

% min_mult_distrib_left
tff(fact_5858_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,Xb: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).

% max_mult_distrib_left
tff(fact_5859_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P2: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),Xb),Y),P2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xb,P2)),divide_divide(A,Y,P2)),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xb,P2)),divide_divide(A,Y,P2))) ) ).

% min_divide_distrib_right
tff(fact_5860_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P2: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),Xb),Y),P2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xb,P2)),divide_divide(A,Y,P2)),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xb,P2)),divide_divide(A,Y,P2))) ) ).

% max_divide_distrib_right
tff(fact_5861_min__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Nb)),Mb) = case_nat(nat,zero_zero(nat),aTP_Lamp_oy(nat,fun(nat,nat),Nb),Mb) ).

% min_Suc1
tff(fact_5862_min__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Mb),aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_oz(nat,fun(nat,nat),Nb),Mb) ).

% min_Suc2
tff(fact_5863_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B5)),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).

% Min.subset_imp
tff(fact_5864_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M6),N3)
         => ( ( M6 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite(A),N3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N3)),aa(set(A),A,lattic643756798350308766er_Min(A),M6)) ) ) ) ) ).

% Min_antimono
tff(fact_5865_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S3: set(A),F3: fun(A,B),K2: B] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_pa(fun(A,B),fun(B,fun(A,B)),F3),K2),S3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798350308766er_Min(B),image(A,B,F3,S3))),K2) ) ) ) ) ).

% Min_add_commute
tff(fact_5866_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Mb: nat,Nb: nat] : modulo_modulo(A,modulo_modulo(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),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)) = 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),ord_min(nat),Mb),Nb))) ) ).

% mod_exp_eq
tff(fact_5867_lexord__take__index__conv,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Y),lexord(A,R2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xb)),aa(list(A),nat,size_size(list(A)),Y))
          & ( take(A,aa(list(A),nat,size_size(list(A)),Xb),Y) = Xb ) )
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xb)),aa(list(A),nat,size_size(list(A)),Y)))
            & ( take(A,I3,Xb) = take(A,I3,Y) )
            & 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,Xb),I3)),aa(nat,A,nth(A,Y),I3)),R2) ) ) ) ).

% lexord_take_index_conv
tff(fact_5868_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_ox(A,fun(A,$o))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_5869_f__arg__min__list__f,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( aa(A,B,F3,arg_min_list(A,B,F3,Xs)) = aa(set(B),B,lattic643756798350308766er_Min(B),image(A,B,F3,aa(list(A),set(A),set2(A),Xs))) ) ) ) ).

% f_arg_min_list_f
tff(fact_5870_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),zero_zero(extended_enat)),Q3) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_5871_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),Q3),zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_5872_lexord__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R2)
     => ~ 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,R2)) ) ).

% lexord_irreflexive
tff(fact_5873_lexord__linear,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: list(A),Y: list(A)] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B4),R2)
          | ( A4 = B4 )
          | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4),R2) )
     => ( 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,R2))
        | ( Xb = Y )
        | 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,R2)) ) ) ).

% lexord_linear
tff(fact_5874_lexord__Nil__right,axiom,
    ! [A: $tType,Xb: list(A),R2: set(product_prod(A,A))] : ~ 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),nil(A)),lexord(A,R2)) ).

% lexord_Nil_right
tff(fact_5875_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X4: A,Y3: A,Z3: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R2)
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R2)
             => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z3),R2) ) ) )
     => ( 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,R2))
       => ( 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),Zs),lexord(A,R2))
         => 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),lexord(A,R2)) ) ) ) ).

% lexord_partial_trans
tff(fact_5876_arg__min__list__in,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => member(A,arg_min_list(A,B,F3,Xs),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% arg_min_list_in
tff(fact_5877_lexord__lex,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( 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,R2))
    <=> ( 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,R2))
        & ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_5878_List_Olexordp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( lexordp(A,R2,Xs,Ys)
    <=> 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),R2)))) ) ).

% List.lexordp_def
tff(fact_5879_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_pb(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys)) ).

% set_zip
tff(fact_5880_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_pc(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Min.eq_fold'
tff(fact_5881_zip__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( zip(A,B,Xs,Ys) = nil(product_prod(A,B)) )
    <=> ( ( Xs = nil(A) )
        | ( Ys = nil(B) ) ) ) ).

% zip_eq_Nil_iff
tff(fact_5882_Nil__eq__zip__iff,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( nil(product_prod(A,B)) = zip(A,B,Xs,Ys) )
    <=> ( ( Xs = nil(A) )
        | ( Ys = nil(B) ) ) ) ).

% Nil_eq_zip_iff
tff(fact_5883_zip__Nil,axiom,
    ! [B: $tType,A: $tType,Ys: list(B)] : zip(A,B,nil(A),Ys) = nil(product_prod(A,B)) ).

% zip_Nil
tff(fact_5884_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I: nat,Xb: A,J: nat,Y: B] : zip(A,B,replicate(A,I,Xb),replicate(B,J,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y)) ).

% zip_replicate
tff(fact_5885_length__zip,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))),zip(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_zip
tff(fact_5886_nth__zip,axiom,
    ! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I)) ) ) ) ).

% nth_zip
tff(fact_5887_inf__enat__def,axiom,
    inf_inf(extended_enat) = ord_min(extended_enat) ).

% inf_enat_def
tff(fact_5888_zip_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Xs: list(A)] : zip(A,B,Xs,nil(B)) = nil(product_prod(A,B)) ).

% zip.simps(1)
tff(fact_5889_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,Xb: A,Ys: list(B),Y: B] : zip(A,B,list_update(A,Xs,I,Xb),list_update(B,Ys,I,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y)) ).

% zip_update
tff(fact_5890_take__zip,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys: list(B)] : take(product_prod(A,B),Nb,zip(A,B,Xs,Ys)) = zip(A,B,take(A,Nb,Xs),take(B,Nb,Ys)) ).

% take_zip
tff(fact_5891_distinct__zipI2,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B)] :
      ( distinct(A,Ys)
     => distinct(product_prod(B,A),zip(B,A,Xs,Ys)) ) ).

% distinct_zipI2
tff(fact_5892_distinct__zipI1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( distinct(A,Xs)
     => distinct(product_prod(A,B),zip(A,B,Xs,Ys)) ) ).

% distinct_zipI1
tff(fact_5893_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ).

% set_zip_rightD
tff(fact_5894_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% set_zip_leftD
tff(fact_5895_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => ~ ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ~ member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ) ).

% in_set_zipE
tff(fact_5896_zip__same,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( 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(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs)))
    <=> ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_5897_update__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),I: nat,Xy: product_prod(A,B)] : list_update(product_prod(A,B),zip(A,B,Xs,Ys),I,Xy) = zip(A,B,list_update(A,Xs,I,aa(product_prod(A,B),A,product_fst(A,B),Xy)),list_update(B,Ys,I,aa(product_prod(A,B),B,product_snd(A,B),Xy))) ).

% update_zip
tff(fact_5898_zip__obtain__same__length,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(product_prod(A,B)),$o)] :
      ( ! [Zs2: list(A),Ws: list(B),N: nat] :
          ( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(B),nat,size_size(list(B)),Ws) )
         => ( ( N = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) )
           => ( ( Zs2 = take(A,N,Xs) )
             => ( ( Ws = take(B,N,Ys) )
               => aa(list(product_prod(A,B)),$o,P,zip(A,B,Zs2,Ws)) ) ) ) )
     => aa(list(product_prod(A,B)),$o,P,zip(A,B,Xs,Ys)) ) ).

% zip_obtain_same_length
tff(fact_5899_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(B,Y,aa(list(B),set(B),set2(B),Ys))
       => ~ ! [X4: A] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_5900_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ~ ! [Y3: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_5901_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A5: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G3),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G3),zero_zero(A),A5) ) ).

% sum.eq_fold
tff(fact_5902_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,times_times(A)),G3),one_one(A),A5) ) ).

% prod.eq_fold
tff(fact_5903_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B),Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),P2,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
    <=> ? [N4: nat] :
          ( ( aa(nat,A,nth(A,Xs),N4) = aa(product_prod(A,B),A,product_fst(A,B),P2) )
          & ( aa(nat,B,nth(B,Ys),N4) = aa(product_prod(A,B),B,product_snd(A,B),P2) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(B),nat,size_size(list(B)),Ys)) ) ) ).

% in_set_zip
tff(fact_5904_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] : lattic7752659483105999362nf_fin(A,A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_pd(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Inf_fin.eq_fold'
tff(fact_5905_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_pe(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Max.eq_fold'
tff(fact_5906_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F3: fun(A,fun(B,B)),Xb: A,A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xb),A5)),S3)
       => ( aa(set(A),$o,finite_finite(A),A5)
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),insert(A,Xb),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xb),finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_5907_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xb)
            <=> ! [X3: A] :
                  ( member(A,X3,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xb) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_5908_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xb)
            <=> ! [X3: A] :
                  ( member(A,X3,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xb) ) ) ) ) ) ).

% Max_less_iff
tff(fact_5909_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ).

% Max.coboundedI
tff(fact_5910_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( ! [X4: A] :
                  ( member(A,X4,A5)
                 => ? [Xa: A] :
                      ( member(A,Xa,B5)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa) ) )
             => ( ! [X4: A] :
                    ( member(A,X4,B5)
                   => ? [Xa: A] :
                        ( member(A,Xa,A5)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Xa) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(set(A),A,lattic643756798349783984er_Max(A),B5) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_5911_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ! [Y3: A] :
                ( member(A,Y3,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
           => ( member(A,Xb,A5)
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = Xb ) ) ) ) ) ).

% Max_eqI
tff(fact_5912_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ).

% Max_ge
tff(fact_5913_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A5)),A2) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_5914_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = Mb )
            <=> ( member(A,Mb,A5)
                & ! [X3: A] :
                    ( member(A,X3,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Mb) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_5915_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A5))
            <=> ? [X3: A] :
                  ( member(A,X3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X3) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_5916_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Mb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( Mb = aa(set(A),A,lattic643756798349783984er_Max(A),A5) )
            <=> ( member(A,Mb,A5)
                & ! [X3: A] :
                    ( member(A,X3,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Mb) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_5917_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xb)
             => ! [A8: A] :
                  ( member(A,A8,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A8),Xb) ) ) ) ) ) ).

% Max.boundedE
tff(fact_5918_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => 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),A5)),Xb) ) ) ) ) ).

% Max.boundedI
tff(fact_5919_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),aa(set(A),A,lattic643756798349783984er_Max(A),A5))
            <=> ? [X3: A] :
                  ( member(A,X3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X3) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_5920_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ! [B4: A] :
                ( member(A,B4,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B4),A2) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,A2),A5)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_5921_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_5922_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic7752659483105999362nf_fin(A,A5))
             => ! [A8: A] :
                  ( member(A,A8,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A8) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_5923_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => 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,A5)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_5924_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic7752659483105999362nf_fin(A,A5))
            <=> ! [X3: A] :
                  ( member(A,X3,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X3) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_5925_card_Oeq__fold,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),nat,finite_card(A),A5) = finite_fold(A,nat,aTP_Lamp_op(A,fun(nat,nat)),zero_zero(nat),A5) ).

% card.eq_fold
tff(fact_5926_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite(A),A5)
         => ( lattic7752659483105999362nf_fin(A,A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_5927_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),nil(A),A5) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_5928_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M6),N3)
         => ( ( M6 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite(A),N3)
             => 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),N3)) ) ) ) ) ).

% Max_mono
tff(fact_5929_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),aa(set(A),A,lattic643756798349783984er_Max(A),B5)) ) ) ) ) ).

% Max.subset_imp
tff(fact_5930_card__le__Suc__Max,axiom,
    ! [S3: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),S3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S3)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S3))) ) ).

% card_le_Suc_Max
tff(fact_5931_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,B5)),lattic7752659483105999362nf_fin(A,A5)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_5932_divide__nat__def,axiom,
    ! [Mb: nat,Nb: 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,$o),aTP_Lamp_pf(nat,fun(nat,fun(nat,$o)),Mb),Nb)))) ).

% divide_nat_def
tff(fact_5933_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S3: set(A),F3: fun(A,B),K2: B] :
          ( aa(set(A),$o,finite_finite(A),S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_pa(fun(A,B),fun(B,fun(A,B)),F3),K2),S3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798349783984er_Max(B),image(A,B,F3,S3))),K2) ) ) ) ) ).

% Max_add_commute
tff(fact_5934_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)
     => ( 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_pg(nat,fun(nat,fun(nat,$o)),Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_5935_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),insert(A,Xb),A5)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_5936_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_5937_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => ( lattic7752659483105999362nf_fin(A,A5) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_5938_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( lattic7752659483105999362nf_fin(A,aa(set(A),set(A),insert(A,Xb),A5)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_5939_sum__le__card__Max,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(A,nat,F3,A5)))) ) ).

% sum_le_card_Max
tff(fact_5940_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S3: set(A),F3: fun(A,fun(B,B)),A5: set(A),Xb: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S3,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S3)
       => ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => ( finite_fold(A,B,F3,Z,A5) = aa(B,B,aa(A,fun(B,B),F3,Xb),finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_5941_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_ox(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_5942_min__list__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( min_list(A,Xs) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% min_list_Min
tff(fact_5943_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_ox(A,fun(A,$o))) = ord_min(A) ) ) ).

% dual_max
tff(fact_5944_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),I: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),aa(nat,A,nth(A,Xs),I)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),I)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_5945_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)),image(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).

% set_concat
tff(fact_5946_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = aa(list(A),list(A),cons(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5)),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_5947_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y22: list(A)] :
      ( ( aa(list(A),list(A),cons(A,X21),X222) = aa(list(A),list(A),cons(A,Y21),Y22) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
tff(fact_5948_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_5949_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_5950_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_5951_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_5952_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(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_5953_nth__Cons__Suc,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ).

% nth_Cons_Suc
tff(fact_5954_nth__Cons__0,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),zero_zero(nat)) = Xb ).

% nth_Cons_0
tff(fact_5955_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),cons(A,Xb),Xs),aa(list(B),list(B),cons(B,Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y)),zip(A,B,Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_5956_take__Suc__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : take(A,aa(nat,nat,suc,Nb),aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(A),list(A),cons(A,Xb),take(A,Nb,Xs)) ).

% take_Suc_Cons
tff(fact_5957_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_5958_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_5959_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_5960_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_5961_map__of__eq__empty__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ! [X3: A] : aa(A,option(B),map_of(A,B,Xys),X3) = none(B)
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% map_of_eq_empty_iff
tff(fact_5962_empty__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ( aTP_Lamp_ph(A,option(B)) = map_of(A,B,Xys) )
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% empty_eq_map_of_iff
tff(fact_5963_nths__singleton,axiom,
    ! [A: $tType,Xb: A,A5: set(nat)] :
      nths(A,aa(list(A),list(A),cons(A,Xb),nil(A)),A5) = $ite(member(nat,zero_zero(nat),A5),aa(list(A),list(A),cons(A,Xb),nil(A)),nil(A)) ).

% nths_singleton
tff(fact_5964_Cons__listrel1__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( 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)),aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(A),list(A),cons(A,Y),Ys)),listrel1(A,R2))
    <=> ( ( 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)
          & ( Xs = Ys ) )
        | ( ( Xb = Y )
          & 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,R2)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_5965_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xb: B,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,aa(list(B),list(B),cons(B,Xb),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F3,Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),groups4207007520872428315er_sum(B,A,F3,A2,Xs))) ) ).

% horner_sum_simps(2)
tff(fact_5966_lexord__cons__cons,axiom,
    ! [A: $tType,A2: A,Xb: list(A),B2: A,Y: list(A),R2: set(product_prod(A,A))] :
      ( 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)),aa(list(A),list(A),cons(A,A2),Xb)),aa(list(A),list(A),cons(A,B2),Y)),lexord(A,R2))
    <=> ( 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)
        | ( ( A2 = B2 )
          & 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,R2)) ) ) ) ).

% lexord_cons_cons
tff(fact_5967_lexord__Nil__left,axiom,
    ! [A: $tType,Y: list(A),R2: set(product_prod(A,A))] :
      ( 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)),nil(A)),Y),lexord(A,R2))
    <=> ? [A6: A,X3: list(A)] : Y = aa(list(A),list(A),cons(A,A6),X3) ) ).

% lexord_Nil_left
tff(fact_5968_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : enumerate(A,Nb,aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),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_5969_n__lists__Nil,axiom,
    ! [A: $tType,Nb: nat] :
      n_lists(A,Nb,nil(A)) = $ite(Nb = zero_zero(nat),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))),nil(list(A))) ).

% n_lists_Nil
tff(fact_5970_nth__Cons__numeral,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(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_5971_take__Cons__numeral,axiom,
    ! [A: $tType,V: num,Xb: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(A),list(A),cons(A,Xb),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs)) ).

% take_Cons_numeral
tff(fact_5972_map__of__zip__is__None,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),Xb) = none(B) )
      <=> ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% map_of_zip_is_None
tff(fact_5973_Cons__in__lex,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( 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)),aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(A),list(A),cons(A,Y),Ys)),lex(A,R2))
    <=> ( ( 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(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( Xb = Y )
          & 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,R2)) ) ) ) ).

% Cons_in_lex
tff(fact_5974_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,aa(list(A),list(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_5975_insort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: 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,F3),Xb),aa(list(A),list(A),cons(A,Y),Ys)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,F3,Y)),aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),cons(A,Y),Ys)),aa(list(A),list(A),cons(A,Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),Ys))) ) ).

% insort_key.simps(2)
tff(fact_5976_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),cons(A,Xb),Xs))
    <=> ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
        & distinct(A,Xs) ) ) ).

% distinct.simps(2)
tff(fact_5977_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( member(A,Y,aa(list(A),set(A),set2(A),X222))
     => member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222))) ) ).

% list.set_intros(2)
tff(fact_5978_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : member(A,X21,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X222))) ).

% list.set_intros(1)
tff(fact_5979_list_Oset__cases,axiom,
    ! [A: $tType,E3: A,A2: list(A)] :
      ( member(A,E3,aa(list(A),set(A),set2(A),A2))
     => ( ! [Z23: list(A)] : A2 != aa(list(A),list(A),cons(A,E3),Z23)
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( A2 = aa(list(A),list(A),cons(A,Z12),Z23) )
             => ~ member(A,E3,aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_5980_set__ConsD,axiom,
    ! [A: $tType,Y: A,Xb: A,Xs: list(A)] :
      ( member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,Xb),Xs)))
     => ( ( Y = Xb )
        | member(A,Y,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% set_ConsD
tff(fact_5981_zip__eq__ConsE,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
      ( ( zip(A,B,Xs,Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),Xy),Xys) )
     => ~ ! [X4: A,Xs4: list(A)] :
            ( ( Xs = aa(list(A),list(A),cons(A,X4),Xs4) )
           => ! [Y3: B,Ys5: list(B)] :
                ( ( Ys = aa(list(B),list(B),cons(B,Y3),Ys5) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) )
                 => ( Xys != zip(A,B,Xs4,Ys5) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_5982_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),aa(list(A),list(A),cons(A,Xb),Xs))) ).

% set_subset_Cons
tff(fact_5983_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Y: A,Xs: list(A),Ys: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),image(list(A),list(A),cons(A,Y),shuffles(A,Xs,Ys))),shuffles(A,Xs,aa(list(A),list(A),cons(A,Y),Ys))) ).

% Cons_shuffles_subset2
tff(fact_5984_Cons__shuffles__subset1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),image(list(A),list(A),cons(A,Xb),shuffles(A,Xs,Ys))),shuffles(A,aa(list(A),list(A),cons(A,Xb),Xs),Ys)) ).

% Cons_shuffles_subset1
tff(fact_5985_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A2) )
           => ( ! [Y3: A] :
                  ( ! [X: A] :
                      ( member(A,X,X6)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_5986_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X6)),Z) ) ) ) ).

% cSup_least
tff(fact_5987_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X6: set(A)] :
          ( member(A,Z,X6)
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_5988_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X6: set(A),A2: A] :
          ( ! [X4: A] :
              ( member(A,X4,X6)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A2) )
         => ( ! [Y3: A] :
                ( ! [X: A] :
                    ( member(A,X,X6)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_5989_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( member(A,Xb,X6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),X6)) ) ) ) ).

% le_cSup_finite
tff(fact_5990_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 != aa(list(A),list(A),cons(A,Xb),Ys) ) ) ).

% impossible_Cons
tff(fact_5991_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_5992_successively_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P5: fun(A,fun(A,$o))] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),nil(A))
     => ( ! [P5: fun(A,fun(A,$o)),X4: A] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),aa(list(A),list(A),cons(A,X4),nil(A)))
       => ~ ! [P5: fun(A,fun(A,$o)),X4: A,Y3: A,Xs2: list(A)] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Xs2))) ) ) ).

% successively.cases
tff(fact_5993_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: product_prod(fun(A,B),list(A))] :
          ( ! [F2: fun(A,B),X4: A] : Xb != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F2),aa(list(A),list(A),cons(A,X4),nil(A)))
         => ( ! [F2: fun(A,B),X4: A,Y3: A,Zs2: list(A)] : Xb != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F2),aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Zs2)))
           => ~ ! [A4: fun(A,B)] : Xb != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A4),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_5994_sorted__wrt_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P5: fun(A,fun(A,$o))] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),nil(A))
     => ~ ! [P5: fun(A,fun(A,$o)),X4: A,Ys3: list(A)] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P5),aa(list(A),list(A),cons(A,X4),Ys3)) ) ).

% sorted_wrt.cases
tff(fact_5995_map__of__Cons__code_I2_J,axiom,
    ! [A: $tType,B: $tType,L: B,V: A,Ps: list(product_prod(B,A)),K2: B] :
      aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),cons(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),L),V)),Ps)),K2) = $ite(L = K2,aa(A,option(A),some(A),V),aa(B,option(A),map_of(B,A,Ps),K2)) ).

% map_of_Cons_code(2)
tff(fact_5996_length__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_5997_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) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,Y5),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_5998_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) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,Y5),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_5999_distinct__length__2__or__more,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),cons(A,A2),aa(list(A),list(A),cons(A,B2),Xs)))
    <=> ( ( A2 != B2 )
        & distinct(A,aa(list(A),list(A),cons(A,A2),Xs))
        & distinct(A,aa(list(A),list(A),cons(A,B2),Xs)) ) ) ).

% distinct_length_2_or_more
tff(fact_6000_not__Cons__self2,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),list(A),cons(A,Xb),Xs) != Xs ).

% not_Cons_self2
tff(fact_6001_remove1_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A)] :
      remove1(A,Xb,aa(list(A),list(A),cons(A,Y),Xs)) = $ite(Xb = Y,Xs,aa(list(A),list(A),cons(A,Y),remove1(A,Xb,Xs))) ).

% remove1.simps(2)
tff(fact_6002_removeAll_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A)] :
      aa(list(A),list(A),removeAll(A,Xb),aa(list(A),list(A),cons(A,Y),Xs)) = $ite(Xb = Y,aa(list(A),list(A),removeAll(A,Xb),Xs),aa(list(A),list(A),cons(A,Y),aa(list(A),list(A),removeAll(A,Xb),Xs))) ).

% removeAll.simps(2)
tff(fact_6003_Cons__in__shuffles__leftI,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z: A] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys))
     => member(list(A),aa(list(A),list(A),cons(A,Z),Zs),shuffles(A,aa(list(A),list(A),cons(A,Z),Xs),Ys)) ) ).

% Cons_in_shuffles_leftI
tff(fact_6004_Cons__in__shuffles__rightI,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z: A] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys))
     => member(list(A),aa(list(A),list(A),cons(A,Z),Zs),shuffles(A,Xs,aa(list(A),list(A),cons(A,Z),Ys))) ) ).

% Cons_in_shuffles_rightI
tff(fact_6005_list_Odistinct_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : nil(A) != aa(list(A),list(A),cons(A,X21),X222) ).

% list.distinct(1)
tff(fact_6006_list_OdiscI,axiom,
    ! [A: $tType,List: list(A),X21: A,X222: list(A)] :
      ( ( List = aa(list(A),list(A),cons(A,X21),X222) )
     => ( List != nil(A) ) ) ).

% list.discI
tff(fact_6007_list_Oexhaust,axiom,
    ! [A: $tType,Y: list(A)] :
      ( ( Y != nil(A) )
     => ~ ! [X212: A,X223: list(A)] : Y != aa(list(A),list(A),cons(A,X212),X223) ) ).

% list.exhaust
tff(fact_6008_min__list_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A)] :
          ( ! [X4: A,Xs2: list(A)] : Xb != aa(list(A),list(A),cons(A,X4),Xs2)
         => ( Xb = nil(A) ) ) ) ).

% min_list.cases
tff(fact_6009_transpose_Ocases,axiom,
    ! [A: $tType,Xb: list(list(A))] :
      ( ( Xb != nil(list(A)) )
     => ( ! [Xss2: list(list(A))] : Xb != aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2)
       => ~ ! [X4: A,Xs2: list(A),Xss2: list(list(A))] : Xb != aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X4),Xs2)),Xss2) ) ) ).

% transpose.cases
tff(fact_6010_remdups__adj_Ocases,axiom,
    ! [A: $tType,Xb: list(A)] :
      ( ( Xb != nil(A) )
     => ( ! [X4: A] : Xb != aa(list(A),list(A),cons(A,X4),nil(A))
       => ~ ! [X4: A,Y3: A,Xs2: list(A)] : Xb != aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_6011_neq__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
    <=> ? [Y5: A,Ys4: list(A)] : Xs = aa(list(A),list(A),cons(A,Y5),Ys4) ) ).

% neq_Nil_conv
tff(fact_6012_list__induct2_H,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
     => ( ! [X4: A,Xs2: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),cons(A,X4),Xs2)),nil(B))
       => ( ! [Y3: B,Ys3: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),aa(list(B),list(B),cons(B,Y3),Ys3))
         => ( ! [X4: A,Xs2: list(A),Y3: B,Ys3: list(B)] :
                ( 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,aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(B),list(B),cons(B,Y3),Ys3)) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_6013_list__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),$o)] :
      ( ( Xs != nil(A) )
     => ( ! [X4: A] : aa(list(A),$o,P,aa(list(A),list(A),cons(A,X4),nil(A)))
       => ( ! [X4: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( aa(list(A),$o,P,Xs2)
               => aa(list(A),$o,P,aa(list(A),list(A),cons(A,X4),Xs2)) ) )
         => aa(list(A),$o,P,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_6014_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))
       => ( ! [X4: A,Xs2: list(A),Y3: 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,aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(B),list(B),cons(B,Y3),Ys3)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_6015_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),$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)),Zs) )
       => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X4: A,Xs2: list(A),Y3: B,Ys3: list(B),Z3: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs2),Ys3),Zs2)
                   => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(B),list(B),cons(B,Y3),Ys3)),aa(list(C),list(C),cons(C,Z3),Zs2)) ) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs),Ys),Zs) ) ) ) ) ).

% list_induct3
tff(fact_6016_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws2: list(D),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D),$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)),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,nil(A)),nil(B)),nil(C)),nil(D))
           => ( ! [X4: A,Xs2: list(A),Y3: B,Ys3: list(B),Z3: C,Zs2: list(C),W2: D,Ws: 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)),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,Xs2),Ys3),Zs2),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,aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(B),list(B),cons(B,Y3),Ys3)),aa(list(C),list(C),cons(C,Z3),Zs2)),aa(list(D),list(D),cons(D,W2),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,Xs),Ys),Zs),Ws2) ) ) ) ) ) ).

% list_induct4
tff(fact_6017_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Xb: A,A2: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( member(A,Xb,X6)
           => ( ! [X4: A] :
                  ( member(A,X4,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),A2) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A2) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_6018_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X6: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ~ ! [X4: A] :
                  ( member(A,X4,X6)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X4) ) ) ) ) ).

% less_cSupE
tff(fact_6019_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X6))
           => ? [X4: A] :
                ( member(A,X4,X6)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X4) ) ) ) ) ).

% less_cSupD
tff(fact_6020_distinct__singleton,axiom,
    ! [A: $tType,Xb: A] : distinct(A,aa(list(A),list(A),cons(A,Xb),nil(A))) ).

% distinct_singleton
tff(fact_6021_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F2: fun(A,B),Bs2: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs2))
     => ~ ! [F2: fun(A,B),A4: A,As: list(A),Bs2: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,A4),As)),Bs2)) ) ).

% map_tailrec_rev.cases
tff(fact_6022_shuffles_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : Xb != 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)),nil(A)),Ys3)
     => ( ! [Xs2: list(A)] : Xb != 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)),Xs2),nil(A))
       => ~ ! [X4: A,Xs2: list(A),Y3: A,Ys3: list(A)] : Xb != 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)),aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(A),list(A),cons(A,Y3),Ys3)) ) ) ).

% shuffles.cases
tff(fact_6023_splice_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : Xb != 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)),nil(A)),Ys3)
     => ~ ! [X4: A,Xs2: list(A),Ys3: list(A)] : Xb != 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)),aa(list(A),list(A),cons(A,X4),Xs2)),Ys3) ) ).

% splice.cases
tff(fact_6024_list__update__code_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] : list_update(A,aa(list(A),list(A),cons(A,Xb),Xs),zero_zero(nat),Y) = aa(list(A),list(A),cons(A,Y),Xs) ).

% list_update_code(2)
tff(fact_6025_list__update__code_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I: nat,Y: A] : list_update(A,aa(list(A),list(A),cons(A,Xb),Xs),aa(nat,nat,suc,I),Y) = aa(list(A),list(A),cons(A,Xb),list_update(A,Xs,I,Y)) ).

% list_update_code(3)
tff(fact_6026_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I: nat,V: A] : list_update(A,aa(list(A),list(A),cons(A,Xb),Xs),I,V) = case_nat(list(A),aa(list(A),list(A),cons(A,V),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_pi(A,fun(list(A),fun(A,fun(nat,list(A)))),Xb),Xs),V),I) ).

% list_update.simps(2)
tff(fact_6027_replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : replicate(A,aa(nat,nat,suc,Nb),Xb) = aa(list(A),list(A),cons(A,Xb),replicate(A,Nb,Xb)) ).

% replicate_Suc
tff(fact_6028_shufflesE,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys))
     => ( ( ( Zs = Xs )
         => ( Ys != nil(A) ) )
       => ( ( ( Zs = Ys )
           => ( Xs != nil(A) ) )
         => ( ! [X4: A,Xs4: list(A)] :
                ( ( Xs = aa(list(A),list(A),cons(A,X4),Xs4) )
               => ! [Z3: A,Zs4: list(A)] :
                    ( ( Zs = aa(list(A),list(A),cons(A,Z3),Zs4) )
                   => ( ( X4 = Z3 )
                     => ~ member(list(A),Zs4,shuffles(A,Xs4,Ys)) ) ) )
           => ~ ! [Y3: A,Ys5: list(A)] :
                  ( ( Ys = aa(list(A),list(A),cons(A,Y3),Ys5) )
                 => ! [Z3: A,Zs4: list(A)] :
                      ( ( Zs = aa(list(A),list(A),cons(A,Z3),Zs4) )
                     => ( ( Y3 = Z3 )
                       => ~ member(list(A),Zs4,shuffles(A,Xs,Ys5)) ) ) ) ) ) ) ) ).

% shufflesE
tff(fact_6029_insort__key_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),nil(A)) = aa(list(A),list(A),cons(A,Xb),nil(A)) ) ).

% insort_key.simps(1)
tff(fact_6030_listrel1I2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Xb: A] :
      ( 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,R2))
     => 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)),aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(A),list(A),cons(A,Xb),Ys)),listrel1(A,R2)) ) ).

% listrel1I2
tff(fact_6031_arg__min__list_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A] : arg_min_list(A,B,F3,aa(list(A),list(A),cons(A,Xb),nil(A))) = Xb ) ).

% arg_min_list.simps(1)
tff(fact_6032_map__of__Cons__code_I1_J,axiom,
    ! [B: $tType,A: $tType,K2: B] : aa(B,option(A),map_of(B,A,nil(product_prod(B,A))),K2) = none(A) ).

% map_of_Cons_code(1)
tff(fact_6033_map__of_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType,X: A] : aa(A,option(B),map_of(A,B,nil(product_prod(A,B))),X) = none(B) ).

% map_of.simps(1)
tff(fact_6034_take__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : take(A,Nb,aa(list(A),list(A),cons(A,Xb),Xs)) = case_nat(list(A),nil(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_pj(A,fun(list(A),fun(nat,list(A))),Xb),Xs),Nb) ).

% take_Cons
tff(fact_6035_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ys: list(A),Xs: list(A)] :
      ( member(list(A),aa(list(A),list(A),cons(A,Y),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))
     => member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) ) ).

% Cons_in_subseqsD
tff(fact_6036_nth__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),Nb) = case_nat(A,Xb,nth(A,Xs),Nb) ).

% nth_Cons
tff(fact_6037_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xb: A,Y: A,Zs: list(A)] :
          arg_min_list(A,B,F3,aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),cons(A,Y),Zs))) = $let(
            m2: A,
            m2:= arg_min_list(A,B,F3,aa(list(A),list(A),cons(A,Y),Zs)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xb)),aa(A,B,F3,m2)),Xb,m2) ) ) ).

% arg_min_list.simps(2)
tff(fact_6038_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),M6: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),M6) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,A5))),M6) ) ) ) ).

% cSUP_least
tff(fact_6039_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A2)
            <=> ! [X3: A] :
                  ( member(A,X3,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),A2) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_6040_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X4)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S3))),A2) ) ) ) ).

% cSup_abs_le
tff(fact_6041_prod_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(set(A)),G3: fun(A,B)] :
          ( ! [X4: set(A)] :
              ( member(set(A),X4,B5)
             => aa(set(A),$o,finite_finite(A),X4) )
         => ( ! [A13: set(A)] :
                ( member(set(A),A13,B5)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B5)
                   => ( ( A13 != A24 )
                     => ! [X4: A] :
                          ( member(A,X4,A13)
                         => ( member(A,X4,A24)
                           => ( aa(A,B,G3,X4) = one_one(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) = 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)),G3),B5) ) ) ) ) ).

% prod.Union_comp
tff(fact_6042_Suc__le__length__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X3: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,X3),Ys4) )
          & aa(nat,$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_6043_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U3: set(set(A))] :
      ( ! [X4: set(A)] :
          ( member(set(A),X4,U3)
         => aa(set(A),$o,finite_finite(A),X4) )
     => 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_6044_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F3: fun(A,B),A2: A] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,A2)),aa(A,B,F3,X4)) )
         => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),A2),Xs) = aa(list(A),list(A),cons(A,A2),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_6045_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( 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),aa(list(A),list(A),cons(A,Y),Ys)),listrel1(A,R2))
     => ( ! [X4: A] :
            ( ( Xs = aa(list(A),list(A),cons(A,X4),Ys) )
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),cons(A,Y),Zs2) )
             => ~ 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)),Zs2),Ys),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E2
tff(fact_6046_Cons__listrel1E1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( 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)),aa(list(A),list(A),cons(A,Xb),Xs)),Ys),listrel1(A,R2))
     => ( ! [Y3: A] :
            ( ( Ys = aa(list(A),list(A),cons(A,Y3),Xs) )
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y3),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),cons(A,Xb),Zs2) )
             => ~ 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),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_6047_listrel1I1,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( 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)
     => 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)),aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(A),list(A),cons(A,Y),Xs)),listrel1(A,R2)) ) ).

% listrel1I1
tff(fact_6048_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K2: A,Xb: B,L: list(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),Xb),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L))
     => ? [X4: B] : aa(A,option(B),map_of(A,B,L),K2) = aa(B,option(B),some(B),X4) ) ).

% weak_map_of_SomeI
tff(fact_6049_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K2: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),Y) )
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),Y),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)) ) ).

% map_of_SomeD
tff(fact_6050_map__of__zip__inject,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xs) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xs) )
       => ( distinct(B,Xs)
         => ( ( map_of(B,A,zip(B,A,Xs,Ys)) = map_of(B,A,zip(B,A,Xs,Zs)) )
           => ( Ys = Zs ) ) ) ) ) ).

% map_of_zip_inject
tff(fact_6051_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] :
      aa(A,nat,count_list(A,aa(list(A),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_6052_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K2: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_pk(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K2),set_ord_lessThan(nat,Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M6,set_or7035219750837199246ssThan(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2)))) ).

% UN_le_add_shift_strict
tff(fact_6053_UN__le__add__shift,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K2: nat,Nb: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_pk(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K2),set_ord_atMost(nat,Nb))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M6,set_or1337092689740270186AtMost(nat,K2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),K2)))) ).

% UN_le_add_shift
tff(fact_6054_map__of__eq__dom,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( map_of(A,B,Xs) = map_of(A,B,Ys) )
     => ( image(product_prod(A,B),A,product_fst(A,B),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)) = image(product_prod(A,B),A,product_fst(A,B),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Ys)) ) ) ).

% map_of_eq_dom
tff(fact_6055_subseqs_Osimps_I1_J,axiom,
    ! [A: $tType] : subseqs(A,nil(A)) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% subseqs.simps(1)
tff(fact_6056_product__lists_Osimps_I1_J,axiom,
    ! [A: $tType] : product_lists(A,nil(list(A))) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% product_lists.simps(1)
tff(fact_6057_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),L))),E3) )
           => aa(A,$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),S3)),L))),E3) ) ) ) ).

% cSup_asclose
tff(fact_6058_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_6059_nth__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(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_6060_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
      <=> ? [Y5: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),Xb) = aa(B,option(B),some(B),Y5) ) ) ).

% map_of_zip_is_Some
tff(fact_6061_map__of__eq__None__iff,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),Xb: B] :
      ( ( aa(B,option(A),map_of(B,A,Xys),Xb) = none(A) )
    <=> ~ member(B,Xb,image(product_prod(B,A),B,product_fst(B,A),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xys))) ) ).

% map_of_eq_None_iff
tff(fact_6062_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X21: A,X222: list(A)] : aa(list(A),nat,size_list(A,Xb),aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X21)),aa(list(A),nat,size_list(A,Xb),X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6063_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_6064_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,I)),J)
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,I)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_6065_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),J)
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,I)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_6066_card__partition,axiom,
    ! [A: $tType,C5: set(set(A)),K2: nat] :
      ( aa(set(set(A)),$o,finite_finite(set(A)),C5)
     => ( aa(set(A),$o,finite_finite(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C5))
       => ( ! [C2: set(A)] :
              ( member(set(A),C2,C5)
             => ( aa(set(A),nat,finite_card(A),C2) = K2 ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( member(set(A),C1,C5)
               => ( 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),K2),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_6067_nth__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$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,aa(list(A),list(A),cons(A,Xb),Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6068_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A),Nb: nat] :
      ( ( Xb != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),Nb) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6069_take__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      take(A,Nb,aa(list(A),list(A),cons(A,Xb),Xs)) = $ite(Nb = zero_zero(nat),nil(A),aa(list(A),list(A),cons(A,Xb),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xs))) ).

% take_Cons'
tff(fact_6070_Cons__replicate__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat,Y: A] :
      ( ( aa(list(A),list(A),cons(A,Xb),Xs) = replicate(A,Nb,Y) )
    <=> ( ( Xb = Y )
        & aa(nat,$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_6071_Cons__lenlex__iff,axiom,
    ! [A: $tType,Mb: A,Ms: list(A),Nb: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,Mb),Ms)),aa(list(A),list(A),cons(A,Nb),Ns)),lenlex(A,R2))
    <=> ( aa(nat,$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) )
          & 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),R2) )
        | ( ( Mb = Nb )
          & 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,R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6072_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite(A),I6)
     => 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)),image(A,set(B),A5,I6)))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_pl(fun(A,set(B)),fun(A,nat),A5)),I6)) ) ).

% card_UN_le
tff(fact_6073_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A5: set(A)] : set_Cons(A,A5,aa(set(list(A)),set(list(A)),insert(list(A),nil(A)),bot_bot(set(list(A))))) = image(A,list(A),aTP_Lamp_pm(A,list(A)),A5) ).

% set_Cons_sing_Nil
tff(fact_6074_SUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ! [X4: A] :
                ( member(A,X4,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,A5))),aa(set(B),B,complete_Sup_Sup(B),image(A,B,G3,B5))) ) ) ) ).

% SUP_subset_mono
tff(fact_6075_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] : aa(A,$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)),A5),B5))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ).

% Sup_inter_less_eq
tff(fact_6076_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X: A,Xa: B] :
      ( aa(B,$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))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S3)) ) ).

% SUP_Sup_eq2
tff(fact_6077_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S3: set(C),X: A,Xa: B] :
      ( aa(B,$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))),image(C,fun(A,fun(B,$o)),aTP_Lamp_pn(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2),S3)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image(C,set(product_prod(A,B)),R2,S3))) ) ).

% SUP_UN_eq2
tff(fact_6078_list__encode_Ocases,axiom,
    ! [Xb: list(nat)] :
      ( ( Xb != nil(nat) )
     => ~ ! [X4: nat,Xs2: list(nat)] : Xb != aa(list(nat),list(nat),cons(nat,X4),Xs2) ) ).

% list_encode.cases
tff(fact_6079_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(fun(A,fun(B,$o))),X: 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,$o))),S3),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),S3)))) ) ).

% Sup_SUP_eq2
tff(fact_6080_listset_Osimps_I2_J,axiom,
    ! [A: $tType,A5: set(A),As2: list(set(A))] : listset(A,aa(list(set(A)),list(set(A)),cons(set(A),A5),As2)) = set_Cons(A,A5,listset(A,As2)) ).

% listset.simps(2)
tff(fact_6081_set__Cons__def,axiom,
    ! [A: $tType,A5: set(A),XS: set(list(A))] : set_Cons(A,A5,XS) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(list(A)),fun(list(A),$o),aTP_Lamp_po(set(A),fun(set(list(A)),fun(list(A),$o)),A5),XS)) ).

% set_Cons_def
tff(fact_6082_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Xb: A] :
          ( ! [Y3: A] :
              ( member(A,Y3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
         => ( ! [Y3: A] :
                ( ! [Z4: A] :
                    ( member(A,Z4,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z4),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A5) = Xb ) ) ) ) ).

% Sup_eqI
tff(fact_6083_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ! [A4: A] :
              ( member(A,A4,A5)
             => ? [X: A] :
                  ( member(A,X,B5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ).

% Sup_mono
tff(fact_6084_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Z: A] :
          ( ! [X4: A] :
              ( member(A,X4,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),Z) ) ) ).

% Sup_least
tff(fact_6085_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A5: set(A)] :
          ( member(A,Xb,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ).

% Sup_upper
tff(fact_6086_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),B2)
        <=> ! [X3: A] :
              ( member(A,X3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B2) ) ) ) ).

% Sup_le_iff
tff(fact_6087_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A5: set(A),V: A] :
          ( member(A,U,A5)
         => ( aa(A,$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),A5)) ) ) ) ).

% Sup_upper2
tff(fact_6088_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S3: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S3))
        <=> ? [X3: A] :
              ( member(A,X3,S3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X3) ) ) ) ).

% less_Sup_iff
tff(fact_6089_Union__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),P: fun(B,$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),$o),aTP_Lamp_pp(fun(B,set(A)),fun(fun(B,$o),fun(set(A),$o)),F3),P))) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_pq(fun(B,set(A)),fun(fun(B,$o),fun(A,$o)),F3),P)) ).

% Union_SetCompr_eq
tff(fact_6090_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,A5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),A5))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xb)
             => ? [X3: A] :
                  ( member(A,X3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X3) ) ) ) ) ).

% le_Sup_iff
tff(fact_6091_SUP__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A5: set(A),B5: set(B),F3: fun(A,C),G3: fun(B,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => ? [X: B] :
                  ( member(B,X,B5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,I2)),aa(B,C,G3,X)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B5)
               => ? [X: A] :
                    ( member(A,X,A5)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G3,J2)),aa(A,C,F3,X)) ) )
           => ( aa(set(C),C,complete_Sup_Sup(C),image(A,C,F3,A5)) = aa(set(C),C,complete_Sup_Sup(C),image(B,C,G3,B5)) ) ) ) ) ).

% SUP_eq
tff(fact_6092_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V3) )
         => ( ( A5 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ).

% less_eq_Sup
tff(fact_6093_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ).

% Sup_subset_mono
tff(fact_6094_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A5: set(A),U: B,F3: fun(A,B)] :
          ( member(A,I,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,I))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,A5))) ) ) ) ).

% SUP_upper2
tff(fact_6095_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F3,A5))),U)
        <=> ! [X3: B] :
              ( member(B,X3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X3)),U) ) ) ) ).

% SUP_le_iff
tff(fact_6096_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,I,A5)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I)),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,A5))) ) ) ).

% SUP_upper
tff(fact_6097_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F3: fun(A,B),G3: fun(A,B),A5: set(A)] :
          ( ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,A5))),aa(set(B),B,complete_Sup_Sup(B),image(A,B,G3,A5))) ) ) ).

% SUP_mono'
tff(fact_6098_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B),U: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),U) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,A5))),U) ) ) ).

% SUP_least
tff(fact_6099_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A5: set(A),B5: set(B),F3: fun(A,C),G3: fun(B,C)] :
          ( ! [N: A] :
              ( member(A,N,A5)
             => ? [X: B] :
                  ( member(B,X,B5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,N)),aa(B,C,G3,X)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),image(A,C,F3,A5))),aa(set(C),C,complete_Sup_Sup(C),image(B,C,G3,B5))) ) ) ).

% SUP_mono
tff(fact_6100_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B),Xb: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),Xb) )
         => ( ! [Y3: B] :
                ( ! [I4: A] :
                    ( member(A,I4,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I4)),Y3) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),Y3) )
           => ( aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,A5)) = Xb ) ) ) ) ).

% SUP_eqI
tff(fact_6101_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),Y: A,I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F3,A5))),Y)
         => ( member(B,I,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,I)),Y) ) ) ) ).

% SUP_lessD
tff(fact_6102_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,F3: fun(B,A),A5: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F3,A5)))
        <=> ? [X3: B] :
              ( member(B,X3,A5)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,X3)) ) ) ) ).

% less_SUP_iff
tff(fact_6103_UN__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),C5: set(B),B5: 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)),image(B,set(A),A5,C5))),B5) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_pr(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5),C5)) ).

% UN_extend_simps(6)
tff(fact_6104_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xb: A,F3: fun(B,A),A5: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F3,A5)))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xb)
             => ? [X3: B] :
                  ( member(B,X3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),aa(B,A,F3,X3)) ) ) ) ) ).

% le_SUP_iff
tff(fact_6105_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I6: set(A),C3: B,F3: fun(A,B)] :
          ( ( I6 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I6)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C3),aa(A,B,F3,I2)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,I6)) = C3 )
            <=> ! [X3: A] :
                  ( member(A,X3,I6)
                 => ( aa(A,B,F3,X3) = C3 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_6106_card__UNION,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite(set(A)),A5)
     => ( ! [X4: set(A)] :
            ( member(set(A),X4,A5)
           => aa(set(A),$o,finite_finite(A),X4) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)) = aa(int,nat,nat2,aa(set(set(set(A))),int,groups7311177749621191930dd_sum(set(set(A)),int,aTP_Lamp_ps(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_pt(set(set(A)),fun(set(set(A)),$o),A5)))) ) ) ) ).

% card_UNION
tff(fact_6107_finite__inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A2: A,A5: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Sup_Sup(A),A5)) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_pu(A,fun(set(A),fun(A,$o)),A2),A5))) ) ).

% finite_inf_Sup
tff(fact_6108_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A5) = bot_bot(A) )
        <=> ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X3)
             => ? [Xa2: A] :
                  ( member(A,Xa2,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X3) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_6109_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_6110_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_6111_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_6112_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_6113_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_6114_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_6115_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_6116_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_6117_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),image(B,A,F3,A5)) = bot_bot(A) )
        <=> ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X3)
             => ? [Xa2: B] :
                  ( member(B,Xa2,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,Xa2)),X3) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_6118_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Xb: A,A2: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( member(A,Xb,X6)
           => ( ! [X4: A] :
                  ( member(A,X4,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_6119_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S3: set(A),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S3)),A2)
        <=> ? [X3: A] :
              ( member(A,X3,S3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),A2) ) ) ) ).

% Inf_less_iff
tff(fact_6120_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Xb: A] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),I2) )
         => ( ! [Y3: A] :
                ( ! [I4: A] :
                    ( member(A,I4,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),I4) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xb) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A5) = Xb ) ) ) ) ).

% Inf_eqI
tff(fact_6121_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( ! [B4: A] :
              ( member(A,B4,B5)
             => ? [X: A] :
                  ( member(A,X,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B4) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ).

% Inf_mono
tff(fact_6122_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xb: A,A5: set(A)] :
          ( member(A,Xb,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),Xb) ) ) ).

% Inf_lower
tff(fact_6123_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A5: set(A),V: A] :
          ( member(A,U,A5)
         => ( aa(A,$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),A5)),V) ) ) ) ).

% Inf_lower2
tff(fact_6124_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A5))
        <=> ! [X3: A] :
              ( member(A,X3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X3) ) ) ) ).

% le_Inf_iff
tff(fact_6125_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Z: A] :
          ( ! [X4: A] :
              ( member(A,X4,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X4) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A5)) ) ) ).

% Inf_greatest
tff(fact_6126_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ).

% Inf_superset_mono
tff(fact_6127_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),Xb)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y5)
             => ? [X3: A] :
                  ( member(A,X3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y5) ) ) ) ) ).

% Inf_le_iff
tff(fact_6128_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( member(A,Xb,X6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Xb) ) ) ) ).

% cInf_le_finite
tff(fact_6129_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X6: set(A),A2: A] :
          ( ! [X4: A] :
              ( member(A,X4,X6)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4) )
         => ( ! [Y3: A] :
                ( ! [X: A] :
                    ( member(A,X,X6)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_6130_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X6: set(A)] :
          ( member(A,Z,X6)
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X4) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_6131_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X4) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),X6)) ) ) ) ).

% cInf_greatest
tff(fact_6132_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,X6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4) )
           => ( ! [Y3: A] :
                  ( ! [X: A] :
                      ( member(A,X,X6)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_6133_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V3),U) )
         => ( ( A5 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),U) ) ) ) ).

% Inf_less_eq
tff(fact_6134_INF__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A5: set(A),B5: set(B),G3: fun(B,C),F3: fun(A,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => ? [X: B] :
                  ( member(B,X,B5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G3,X)),aa(A,C,F3,I2)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B5)
               => ? [X: A] :
                    ( member(A,X,A5)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,X)),aa(B,C,G3,J2)) ) )
           => ( aa(set(C),C,complete_Inf_Inf(C),image(A,C,F3,A5)) = aa(set(C),C,complete_Inf_Inf(C),image(B,C,G3,B5)) ) ) ) ) ).

% INF_eq
tff(fact_6135_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Z)
           => ? [X4: A] :
                ( member(A,X4,X6)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Z) ) ) ) ) ).

% cInf_lessD
tff(fact_6136_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),Xb: B,F3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),aa(A,B,F3,I2)) )
         => ( ! [Y3: B] :
                ( ! [I4: A] :
                    ( member(A,I4,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y3),aa(A,B,F3,I4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y3),Xb) )
           => ( aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,A5)) = Xb ) ) ) ) ).

% INF_eqI
tff(fact_6137_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B5: set(A),A5: set(B),F3: fun(B,C),G3: fun(A,C)] :
          ( ! [M: A] :
              ( member(A,M,B5)
             => ? [X: B] :
                  ( member(B,X,A5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F3,X)),aa(A,C,G3,M)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),image(B,C,F3,A5))),aa(set(C),C,complete_Inf_Inf(C),image(A,C,G3,B5))) ) ) ).

% INF_mono
tff(fact_6138_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,I,A5)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,A5))),aa(A,B,F3,I)) ) ) ).

% INF_lower
tff(fact_6139_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F3: fun(A,B),G3: fun(A,B),A5: set(A)] :
          ( ! [X4: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,A5))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,G3,A5))) ) ) ).

% INF_mono'
tff(fact_6140_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A5: set(A),F3: fun(A,B),U: B] :
          ( member(A,I,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,A5))),U) ) ) ) ).

% INF_lower2
tff(fact_6141_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F3: fun(B,A),A5: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F3,A5)))
        <=> ! [X3: B] :
              ( member(B,X3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F3,X3)) ) ) ) ).

% le_INF_iff
tff(fact_6142_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),U: B,F3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,A5))) ) ) ).

% INF_greatest
tff(fact_6143_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F3,A5))),A2)
        <=> ? [X3: B] :
              ( member(B,X3,A5)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X3)),A2) ) ) ) ).

% INF_less_iff
tff(fact_6144_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F3: fun(B,A),A5: set(B),I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F3,A5)))
         => ( member(B,I,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F3,I)) ) ) ) ).

% less_INF_D
tff(fact_6145_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B),Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F3,A5))),Xb)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y5)
             => ? [X3: B] :
                  ( member(B,X3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X3)),Y5) ) ) ) ) ).

% INF_le_iff
tff(fact_6146_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),Mb: B,F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(A,B,F3,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Mb),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,A5))) ) ) ) ).

% cINF_greatest
tff(fact_6147_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I6: set(A),F3: fun(A,B),C3: B] :
          ( ( I6 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I6)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),C3) )
           => ( ( aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,I6)) = C3 )
            <=> ! [X3: A] :
                  ( member(A,X3,I6)
                 => ( aa(A,B,F3,X3) = C3 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_6148_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),X6)
         => ( ( X6 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6))
            <=> ! [X3: A] :
                  ( member(A,X3,X6)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X3) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_6149_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ).

% Inf_le_Sup
tff(fact_6150_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),A2: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X4)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S3))),A2) ) ) ) ).

% cInf_abs_ge
tff(fact_6151_INF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( ! [X4: A] :
                ( member(A,X4,B5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,A5))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,G3,B5))) ) ) ) ).

% INF_superset_mono
tff(fact_6152_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A5) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_pv(set(A),fun(A,$o),A5))) ) ).

% Inf_eq_Sup
tff(fact_6153_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A5) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_pw(set(A),fun(A,$o),A5))) ) ).

% Sup_eq_Inf
tff(fact_6154_UN__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),B5,C5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_px(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5),C5)) ).

% UN_extend_simps(7)
tff(fact_6155_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,A5))),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,A5))) ) ) ).

% INF_le_SUP
tff(fact_6156_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S3: set(A),L: A,E3: A] :
          ( ( S3 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( member(A,X4,S3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),L))),E3) )
           => aa(A,$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),S3)),L))),E3) ) ) ) ).

% cInf_asclose
tff(fact_6157_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C5: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),B5,C5))) = $ite(C5 = bot_bot(set(B)),A5,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_px(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5),C5))) ).

% INT_extend_simps(4)
tff(fact_6158_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: A,B5: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),aa(set(A),A,complete_Inf_Inf(A),image(nat,A,aTP_Lamp_py(A,fun(nat,A),B5),aa(fun(nat,$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),A5),B5) ) ).

% INF_nat_binary
tff(fact_6159_list__encode_Oelims,axiom,
    ! [Xb: list(nat),Y: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,Xb) = Y )
     => ( ( ( Xb = nil(nat) )
         => ( Y != zero_zero(nat) ) )
       => ~ ! [X4: nat,Xs2: list(nat)] :
              ( ( Xb = aa(list(nat),list(nat),cons(nat,X4),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),X4),aa(list(nat),nat,nat_list_encode,Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_6160_Inf__real__def,axiom,
    ! [X6: set(real)] : aa(set(real),real,complete_Inf_Inf(real),X6) = aa(real,real,uminus_uminus(real),aa(set(real),real,complete_Sup_Sup(real),image(real,real,uminus_uminus(real),X6))) ).

% Inf_real_def
tff(fact_6161_list__encode__eq,axiom,
    ! [Xb: list(nat),Y: list(nat)] :
      ( ( aa(list(nat),nat,nat_list_encode,Xb) = aa(list(nat),nat,nat_list_encode,Y) )
    <=> ( Xb = Y ) ) ).

% list_encode_eq
tff(fact_6162_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X: A,Xa: B] :
      ( aa(B,$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))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S3)) ) ).

% INF_Int_eq2
tff(fact_6163_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S3: set(C),X: A,Xa: B] :
      ( aa(B,$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))),image(C,fun(A,fun(B,$o)),aTP_Lamp_pn(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2),S3)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),image(C,set(product_prod(A,B)),R2,S3))) ) ).

% INF_INT_eq2
tff(fact_6164_list__encode_Osimps_I1_J,axiom,
    aa(list(nat),nat,nat_list_encode,nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_6165_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X3: product_prod(A,A)] :
            ( member(product_prod(A,A),X3,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys)))
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),fequal(A)),X3) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_6166_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A5: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),A5))),aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_pz(set(set(A)),fun(set(A),$o),A5))))) ) ).

% Inf_Sup_le
tff(fact_6167_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_qa(set(set(A)),fun(set(A),$o),A5))))),aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),A5))) ) ).

% Sup_Inf_le
tff(fact_6168_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A5: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),A5))),aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_qb(set(set(A)),fun(set(A),$o),A5))))) ) ).

% finite_Inf_Sup
tff(fact_6169_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ! [X4: product_prod(list(A),list(A))] :
          ( member(product_prod(list(A),list(A)),X4,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys)))
         => aa(product_prod(list(A),list(A)),$o,aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_qc(list(A),fun(list(A),$o))),X4) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ( concat(A,Xs) = concat(A,Ys) )
        <=> ( Xs = Ys ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_6170_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ! [X4: product_prod(list(A),list(A))] :
              ( member(product_prod(list(A),list(A)),X4,aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys)))
             => aa(product_prod(list(A),list(A)),$o,aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_qc(list(A),fun(list(A),$o))),X4) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_6171_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S3: set(fun(A,fun(B,$o))),X: 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,$o))),S3),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),S3)))) ) ).

% Inf_INT_eq2
tff(fact_6172_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list(nat)] : aa(list(nat),nat,nat_list_encode,aa(list(nat),list(nat),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),aa(list(nat),nat,nat_list_encode,Xs)))) ).

% list_encode.simps(2)
tff(fact_6173_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Xb),nil(A))),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% concat_inth
tff(fact_6174_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)),image(list(A),set(A),set2(A),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))) ).

% length_remdups_concat
tff(fact_6175_same__append__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs) )
    <=> ( Ys = Zs ) ) ).

% same_append_eq
tff(fact_6176_append__same__eq,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Xs) )
    <=> ( Ys = Zs ) ) ).

% append_same_eq
tff(fact_6177_append__assoc,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Zs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs)) ).

% append_assoc
tff(fact_6178_append_Oassoc,axiom,
    ! [A: $tType,A2: list(A),B2: list(A),C3: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),B2)),C3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),B2),C3)) ).

% append.assoc
tff(fact_6179_append__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = nil(A) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% append_is_Nil_conv
tff(fact_6180_Nil__is__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_is_append_conv
tff(fact_6181_self__append__conv2,axiom,
    ! [A: $tType,Y: list(A),Xs: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Y) )
    <=> ( Xs = nil(A) ) ) ).

% self_append_conv2
tff(fact_6182_append__self__conv2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Ys )
    <=> ( Xs = nil(A) ) ) ).

% append_self_conv2
tff(fact_6183_self__append__conv,axiom,
    ! [A: $tType,Y: list(A),Ys: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Y),Ys) )
    <=> ( Ys = nil(A) ) ) ).

% self_append_conv
tff(fact_6184_append__self__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Xs )
    <=> ( Ys = nil(A) ) ) ).

% append_self_conv
tff(fact_6185_append__Nil2,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),nil(A)) = Xs ).

% append_Nil2
tff(fact_6186_append_Oright__neutral,axiom,
    ! [A: $tType,A2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),nil(A)) = A2 ).

% append.right_neutral
tff(fact_6187_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) ) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_6188_concat__append,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xs)),concat(A,Ys)) ).

% concat_append
tff(fact_6189_remdups__eq__nil__right__iff,axiom,
    ! [A: $tType,Xb: list(A)] :
      ( ( nil(A) = remdups(A,Xb) )
    <=> ( Xb = nil(A) ) ) ).

% remdups_eq_nil_right_iff
tff(fact_6190_remdups__eq__nil__iff,axiom,
    ! [A: $tType,Xb: list(A)] :
      ( ( remdups(A,Xb) = nil(A) )
    <=> ( Xb = nil(A) ) ) ).

% remdups_eq_nil_iff
tff(fact_6191_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_6192_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_6193_distinct__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : distinct(A,remdups(A,Xs)) ).

% distinct_remdups
tff(fact_6194_remdups__id__iff__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( remdups(A,Xs) = Xs )
    <=> distinct(A,Xs) ) ).

% remdups_id_iff_distinct
tff(fact_6195_removeAll__append,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),removeAll(A,Xb),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),removeAll(A,Xb),Xs)),aa(list(A),list(A),removeAll(A,Xb),Ys)) ).

% removeAll_append
tff(fact_6196_append1__eq__conv,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),cons(A,Y),nil(A))) )
    <=> ( ( Xs = Ys )
        & ( Xb = Y ) ) ) ).

% append1_eq_conv
tff(fact_6197_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),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_6198_zip__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Us: list(B),Ys: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Us) )
     => ( zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us),Vs)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),zip(A,B,Xs,Us)),zip(A,B,Ys,Vs)) ) ) ).

% zip_append
tff(fact_6199_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_6200_size__list__append,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_list(A,F3),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_list(A,F3),Xs)),aa(list(A),nat,size_list(A,F3),Ys)) ).

% size_list_append
tff(fact_6201_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% nth_append_length
tff(fact_6202_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),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)),Nb)) = aa(nat,A,nth(A,Ys),Nb) ).

% nth_append_length_plus
tff(fact_6203_take__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : take(A,Nb,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Nb,Xs)),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% take_append
tff(fact_6204_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A] : list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),Ys)),aa(list(A),nat,size_size(list(A)),Xs),Y) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Y),Ys)) ).

% list_update_length
tff(fact_6205_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(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_6206_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_lessThan(nat,aa(nat,nat,suc,K2))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_lessThan(nat,K2))),aa(list(nat),list(nat),cons(nat,K2),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_6207_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_atMost(nat,aa(nat,nat,suc,K2))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_ord_atMost(nat,K2))),aa(list(nat),list(nat),cons(nat,aa(nat,nat,suc,K2)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_6208_lex__append__leftI,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),R2: set(product_prod(A,A)),Xs: list(A)] :
      ( 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),Zs),lex(A,R2))
     => 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lex(A,R2)) ) ).

% lex_append_leftI
tff(fact_6209_append__listrel1I,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Us: list(A),Vs: list(A)] :
      ( ( ( 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,R2))
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & 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),Vs),listrel1(A,R2)) ) )
     => 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs)),listrel1(A,R2)) ) ).

% append_listrel1I
tff(fact_6210_lexord__append__leftI,axiom,
    ! [A: $tType,U: list(A),V: list(A),R2: set(product_prod(A,A)),Xb: list(A)] :
      ( 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,R2))
     => 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),V)),lexord(A,R2)) ) ).

% lexord_append_leftI
tff(fact_6211_remove1__append,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] :
      remove1(A,Xb,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remove1(A,Xb,Xs)),Ys),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),remove1(A,Xb,Ys))) ).

% remove1_append
tff(fact_6212_distinct__remdups__id,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups(A,Xs) = Xs ) ) ).

% distinct_remdups_id
tff(fact_6213_enumerate__append__eq,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : enumerate(A,Nb,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(list(product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),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_6214_append__eq__append__conv2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),Ts2: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Ts2) )
    <=> ? [Us2: list(A)] :
          ( ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us2) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ys) = Ts2 ) )
          | ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2) = Zs )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ts2) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_6215_append__eq__appendI,axiom,
    ! [A: $tType,Xs: list(A),Xs1: list(A),Zs: list(A),Ys: list(A),Us: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Xs1) = Zs )
     => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Us) )
       => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us) ) ) ) ).

% append_eq_appendI
tff(fact_6216_remdups__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : remdups(A,remdups(A,Xs)) = remdups(A,Xs) ).

% remdups_remdups
tff(fact_6217_remdups__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : remdups(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),remdups(A,Ys))) = remdups(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% remdups_append2
tff(fact_6218_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) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xb)),replicate(A,Mb,Xb)) ).

% replicate_add
tff(fact_6219_append__replicate__commute,axiom,
    ! [A: $tType,Nb: nat,Xb: A,K2: nat] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xb)),replicate(A,K2,Xb)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,K2,Xb)),replicate(A,Nb,Xb)) ).

% append_replicate_commute
tff(fact_6220_concat_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: list(A),Xs: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),cons(list(A),Xb),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),concat(A,Xs)) ).

% concat.simps(2)
tff(fact_6221_eq__Nil__appendI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_6222_append_Oleft__neutral,axiom,
    ! [A: $tType,A2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),A2) = A2 ).

% append.left_neutral
tff(fact_6223_append__Nil,axiom,
    ! [A: $tType,Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) = Ys ).

% append_Nil
tff(fact_6224_remdups_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups(A,nil(A)) = nil(A) ).

% remdups.simps(1)
tff(fact_6225_replicate__app__Cons__same,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xb)),aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Nb,Xb)),Xs)) ).

% replicate_app_Cons_same
tff(fact_6226_concat__eq__appendD,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs: list(A)] :
      ( ( concat(A,Xss) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
     => ( ( Xss != nil(list(A)) )
       => ? [Xss1: list(list(A)),Xs2: list(A),Xs4: list(A),Xss22: list(list(A))] :
            ( ( Xss = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss1),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),Xs4)),Xss22)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xss1)),Xs2) )
            & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),concat(A,Xss22)) ) ) ) ) ).

% concat_eq_appendD
tff(fact_6227_concat__eq__append__conv,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs: list(A)] :
      ( ( concat(A,Xss) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> $ite(
          Xss = nil(list(A)),
          ( ( Ys = nil(A) )
          & ( Zs = nil(A) ) ),
          ? [Xss12: list(list(A)),Xs3: list(A),Xs5: list(A),Xss23: list(list(A))] :
            ( ( Xss = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss12),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),Xs5)),Xss23)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xss12)),Xs3) )
            & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs5),concat(A,Xss23)) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_6228_rev__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),$o)] :
      ( ( Xs != nil(A) )
     => ( ! [X4: A] : aa(list(A),$o,P,aa(list(A),list(A),cons(A,X4),nil(A)))
       => ( ! [X4: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( aa(list(A),$o,P,Xs2)
               => aa(list(A),$o,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),cons(A,X4),nil(A)))) ) )
         => aa(list(A),$o,P,Xs) ) ) ) ).

% rev_nonempty_induct
tff(fact_6229_append__eq__Cons__conv,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),cons(A,Xb),Xs) )
    <=> ( ( ( Ys = nil(A) )
          & ( Zs = aa(list(A),list(A),cons(A,Xb),Xs) ) )
        | ? [Ys6: list(A)] :
            ( ( Ys = aa(list(A),list(A),cons(A,Xb),Ys6) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_6230_Cons__eq__append__conv,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),cons(A,Xb),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ( ( ( Ys = nil(A) )
          & ( aa(list(A),list(A),cons(A,Xb),Xs) = Zs ) )
        | ? [Ys6: list(A)] :
            ( ( aa(list(A),list(A),cons(A,Xb),Ys6) = Ys )
            & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_6231_rev__exhaust,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ~ ! [Ys3: list(A),Y3: A] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,Y3),nil(A))) ) ).

% rev_exhaust
tff(fact_6232_rev__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [X4: A,Xs2: list(A)] :
            ( aa(list(A),$o,P,Xs2)
           => aa(list(A),$o,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),cons(A,X4),nil(A)))) )
       => aa(list(A),$o,P,Xs) ) ) ).

% rev_induct
tff(fact_6233_append__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Xb),Xs)),Ys) = aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% append_Cons
tff(fact_6234_Cons__eq__appendI,axiom,
    ! [A: $tType,Xb: A,Xs1: list(A),Ys: list(A),Xs: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),cons(A,Xb),Xs1) = Ys )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Zs) )
       => ( aa(list(A),list(A),cons(A,Xb),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) ) ) ) ).

% Cons_eq_appendI
tff(fact_6235_split__list,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys3: list(A),Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,Xb),Zs2)) ) ).

% split_list
tff(fact_6236_split__list__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys3: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,Xb),Zs2)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Zs2)) ) ) ).

% split_list_last
tff(fact_6237_split__list__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ? [Ys3: list(A),X4: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,X4),Zs2))
          & aa(A,$o,P,X4) ) ) ).

% split_list_prop
tff(fact_6238_split__list__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys3: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,Xb),Zs2)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys3)) ) ) ).

% split_list_first
tff(fact_6239_split__list__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ~ ! [Ys3: list(A),X4: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,X4),Zs2))
           => ~ aa(A,$o,P,X4) ) ) ).

% split_list_propE
tff(fact_6240_append__Cons__eq__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),Xs6: list(A),Ys7: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys))
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs6),aa(list(A),list(A),cons(A,Xb),Ys7)) )
        <=> ( ( Xs = Xs6 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_6241_in__set__conv__decomp,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys4: list(A),Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,Xb),Zs3)) ) ).

% in_set_conv_decomp
tff(fact_6242_split__list__last__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ? [Ys3: list(A),X4: A,Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,X4),Zs2)) )
          & aa(A,$o,P,X4)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_last_prop
tff(fact_6243_split__list__first__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ? [Ys3: list(A),X4: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,X4),Zs2))
          & aa(A,$o,P,X4)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_first_prop
tff(fact_6244_split__list__last__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ~ ! [Ys3: list(A),X4: A,Zs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,X4),Zs2)) )
           => ( aa(A,$o,P,X4)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_last_propE
tff(fact_6245_split__list__first__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
     => ~ ! [Ys3: list(A),X4: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),cons(A,X4),Zs2))
           => ( aa(A,$o,P,X4)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_first_propE
tff(fact_6246_in__set__conv__decomp__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,Xb),Zs3)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Zs3)) ) ) ).

% in_set_conv_decomp_last
tff(fact_6247_in__set__conv__decomp__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,Xb),Zs3)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys4)) ) ) ).

% in_set_conv_decomp_first
tff(fact_6248_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
    <=> ? [Ys4: list(A),X3: A,Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,X3),Zs3)) )
          & aa(A,$o,P,X3)
          & ! [Xa2: A] :
              ( member(A,Xa2,aa(list(A),set(A),set2(A),Zs3))
             => ~ aa(A,$o,P,Xa2) ) ) ) ).

% split_list_last_prop_iff
tff(fact_6249_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
    <=> ? [Ys4: list(A),X3: A] :
          ( ? [Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,X3),Zs3))
          & aa(A,$o,P,X3)
          & ! [Xa2: A] :
              ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys4))
             => ~ aa(A,$o,P,Xa2) ) ) ) ).

% split_list_first_prop_iff
tff(fact_6250_remdups_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      remdups(A,aa(list(A),list(A),cons(A,Xb),Xs)) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),Xs)),remdups(A,Xs),aa(list(A),list(A),cons(A,Xb),remdups(A,Xs))) ).

% remdups.simps(2)
tff(fact_6251_remove1__remdups,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( remove1(A,Xb,remdups(A,Xs)) = remdups(A,remove1(A,Xb,Xs)) ) ) ).

% remove1_remdups
tff(fact_6252_comm__append__are__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
     => ? [M: nat,N: nat,Zs2: list(A)] :
          ( ( concat(A,replicate(list(A),M,Zs2)) = Xs )
          & ( concat(A,replicate(list(A),N,Zs2)) = Ys ) ) ) ).

% comm_append_are_replicate
tff(fact_6253_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),X4: A,Xs4: list(A),Y3: A,Ys5: list(A)] :
            ( ( X4 != Y3 )
            & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,X4),nil(A))),Xs4)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Y3),nil(A))),Ys5)) ) ) ) ) ).

% same_length_different
tff(fact_6254_not__distinct__decomp,axiom,
    ! [A: $tType,Ws2: list(A)] :
      ( ~ distinct(A,Ws2)
     => ? [Xs2: list(A),Ys3: list(A),Zs2: list(A),Y3: A] : Ws2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Y3),nil(A))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Y3),nil(A))),Zs2)))) ) ).

% not_distinct_decomp
tff(fact_6255_not__distinct__conv__prefix,axiom,
    ! [A: $tType,As3: list(A)] :
      ( ~ distinct(A,As3)
    <=> ? [Xs3: list(A),Y5: A,Ys4: list(A)] :
          ( member(A,Y5,aa(list(A),set(A),set2(A),Xs3))
          & distinct(A,Xs3)
          & ( As3 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),cons(A,Y5),Ys4)) ) ) ) ).

% not_distinct_conv_prefix
tff(fact_6256_replicate__append__same,axiom,
    ! [A: $tType,I: nat,Xb: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,I,Xb)),aa(list(A),list(A),cons(A,Xb),nil(A))) = aa(list(A),list(A),cons(A,Xb),replicate(A,I,Xb)) ).

% replicate_append_same
tff(fact_6257_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Ys: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),I,Xb) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,I,Xb)),Ys) ) ) ).

% list_update_append1
tff(fact_6258_remove1__split,axiom,
    ! [A: $tType,A2: A,Xs: list(A),Ys: list(A)] :
      ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
     => ( ( remove1(A,A2,Xs) = Ys )
      <=> ? [Ls: list(A),Rs: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls),aa(list(A),list(A),cons(A,A2),Rs)) )
            & ~ member(A,A2,aa(list(A),set(A),set2(A),Ls))
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls),Rs) ) ) ) ) ).

% remove1_split
tff(fact_6259_lexord__append__leftD,axiom,
    ! [A: $tType,Xb: list(A),U: list(A),V: list(A),R2: set(product_prod(A,A))] :
      ( 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),V)),lexord(A,R2))
     => ( ! [A4: A] : ~ 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),R2)
       => 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,R2)) ) ) ).

% lexord_append_leftD
tff(fact_6260_lexord__append__rightI,axiom,
    ! [A: $tType,Y: list(A),Xb: list(A),R2: set(product_prod(A,A))] :
      ( ? [B9: A,Z4: list(A)] : Y = aa(list(A),list(A),cons(A,B9),Z4)
     => 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),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xb),Y)),lexord(A,R2)) ) ).

% lexord_append_rightI
tff(fact_6261_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R2: set(product_prod(A,A))] :
      ( 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Qs)),lexord(A,R2))
     => ( ( 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)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
           => 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,R2)) ) ) ) ) ).

% lexord_sufE
tff(fact_6262_lex__append__left__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R2)
     => ( 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lex(A,R2))
      <=> 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),Zs),lex(A,R2)) ) ) ).

% lex_append_left_iff
tff(fact_6263_lex__append__leftD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R2)
     => ( 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lex(A,R2))
       => 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),Zs),lex(A,R2)) ) ) ).

% lex_append_leftD
tff(fact_6264_rotate1_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),list(A),rotate1(A),aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),nil(A))) ).

% rotate1.simps(2)
tff(fact_6265_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( 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,R2))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs)),lex(A,R2)) ) ) ).

% lex_append_rightI
tff(fact_6266_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
      ( 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,R))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Vs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),lenlex(A,R)) ) ) ).

% lenlex_append1
tff(fact_6267_nths__append,axiom,
    ! [A: $tType,L: list(A),L2: list(A),A5: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L2),A5) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A5)),nths(A,L2,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_qd(list(A),fun(set(nat),fun(nat,$o)),L),A5)))) ).

% nths_append
tff(fact_6268_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_6269_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) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),cons(A,Y5),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = Nb ) ) ) ).

% length_Suc_conv_rev
tff(fact_6270_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_6271_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(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_6272_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat,Xb: A] :
      list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,Nb,Xb)),Ys),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),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_6273_listrel1I,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( 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)
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),cons(A,Xb),Vs)) )
       => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),cons(A,Y),Vs)) )
         => 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,R2)) ) ) ) ).

% listrel1I
tff(fact_6274_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( 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,R2))
     => ~ ! [X4: A,Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R2)
           => ! [Us3: list(A),Vs2: list(A)] :
                ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),cons(A,X4),Vs2)) )
               => ( Ys != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),cons(A,Y3),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_6275_lexord__append__left__rightI,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),U: list(A),Xb: list(A),Y: list(A)] :
      ( 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)
     => 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),cons(A,A2),Xb))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),cons(A,B2),Y))),lexord(A,R2)) ) ).

% lexord_append_left_rightI
tff(fact_6276_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R2: set(product_prod(A,A))] :
      ( 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lexord(A,R2))
    <=> ( ? [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2) )
        | 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),Zs),lexord(A,R2)) ) ) ).

% lexord_same_pref_iff
tff(fact_6277_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W: list(A),R2: set(product_prod(A,A)),V: list(A),Z: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W),lexord(A,R2))
     => ( aa(nat,$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))
       => 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),Z)),lexord(A,R2)) ) ) ).

% lexord_sufI
tff(fact_6278_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A,R2: set(product_prod(A,A))] :
      ( 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),cons(A,Y),nil(A)))),listrel1(A,R2))
    <=> ( ( 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,R2))
          & ( Xb = Y ) )
        | ( ( Xs = Ys )
          & 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) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_6279_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups4207007520872428315er_sum(B,A,F3,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,F3,A2,Ys))) ) ).

% horner_sum_append
tff(fact_6280_nths__Cons,axiom,
    ! [A: $tType,Xb: A,L: list(A),A5: set(nat)] :
      nths(A,aa(list(A),list(A),cons(A,Xb),L),A5) = aa(list(A),list(A),
        aa(list(A),fun(list(A),list(A)),append(A),
          $ite(member(nat,zero_zero(nat),A5),aa(list(A),list(A),cons(A,Xb),nil(A)),nil(A))),
        nths(A,L,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_qe(set(nat),fun(nat,$o),A5)))) ).

% nths_Cons
tff(fact_6281_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
         => ? [N: nat,Zs2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N)
              & ( concat(A,replicate(list(A),N,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_6282_listrel1__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,R2) = aa(fun(product_prod(list(A),list(A)),$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),$o),aTP_Lamp_qf(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% listrel1_def
tff(fact_6283_lexord__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lexord(A,R2) = aa(fun(product_prod(list(A),list(A)),$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),$o),aTP_Lamp_qg(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lexord_def
tff(fact_6284_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,aa(nat,nat,suc,I),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I,Xs)),aa(list(A),list(A),cons(A,aa(nat,A,nth(A,Xs),I)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_6285_lex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = aa(fun(product_prod(list(A),list(A)),$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),$o),aTP_Lamp_qh(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lex_conv
tff(fact_6286_nth__repl,axiom,
    ! [A: $tType,Mb: nat,Xs: list(A),Nb: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( Mb != Nb )
         => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Nb,Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Xb),nil(A))),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),Xs)))),Mb) = aa(nat,A,nth(A,Xs),Mb) ) ) ) ) ).

% nth_repl
tff(fact_6287_pos__n__replace,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Nb,Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),cons(A,Y),nil(A))),drop(A,aa(nat,nat,suc,Nb),Xs)))) ) ) ).

% pos_n_replace
tff(fact_6288_drop0,axiom,
    ! [A: $tType,X: list(A)] : drop(A,zero_zero(nat),X) = X ).

% drop0
tff(fact_6289_drop__drop,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] : drop(A,Nb,drop(A,Mb,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb),Xs) ).

% drop_drop
tff(fact_6290_drop__Suc__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : drop(A,aa(nat,nat,suc,Nb),aa(list(A),list(A),cons(A,Xb),Xs)) = drop(A,Nb,Xs) ).

% drop_Suc_Cons
tff(fact_6291_length__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),drop(A,Nb,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ).

% length_drop
tff(fact_6292_drop__update__cancel,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),Mb)
     => ( drop(A,Mb,list_update(A,Xs,Nb,Xb)) = drop(A,Mb,Xs) ) ) ).

% drop_update_cancel
tff(fact_6293_append__take__drop__id,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Nb,Xs)),drop(A,Nb,Xs)) = Xs ).

% append_take_drop_id
tff(fact_6294_drop__replicate,axiom,
    ! [A: $tType,I: nat,K2: nat,Xb: A] : drop(A,I,replicate(A,K2,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K2),I),Xb) ).

% drop_replicate
tff(fact_6295_drop__eq__Nil2,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,Nb,Xs) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% drop_eq_Nil2
tff(fact_6296_drop__eq__Nil,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( drop(A,Nb,Xs) = nil(A) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ) ).

% drop_eq_Nil
tff(fact_6297_drop__all,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb)
     => ( drop(A,Nb,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_6298_drop__append,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : drop(A,Nb,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,Nb,Xs)),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% drop_append
tff(fact_6299_drop__Cons__numeral,axiom,
    ! [A: $tType,V: num,Xb: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),cons(A,Xb),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs) ).

% drop_Cons_numeral
tff(fact_6300_nth__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),I: nat] :
      ( aa(nat,$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,drop(A,Nb,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),I)) ) ) ).

% nth_drop
tff(fact_6301_drop__zip,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys: list(B)] : drop(product_prod(A,B),Nb,zip(A,B,Xs,Ys)) = zip(A,B,drop(A,Nb,Xs),drop(B,Nb,Ys)) ).

% drop_zip
tff(fact_6302_in__set__dropD,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),drop(A,Nb,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_dropD
tff(fact_6303_set__drop__subset,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,Nb,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_drop_subset
tff(fact_6304_nth__via__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Y: A,Ys: list(A)] :
      ( ( drop(A,Nb,Xs) = aa(list(A),list(A),cons(A,Y),Ys) )
     => ( aa(nat,A,nth(A,Xs),Nb) = Y ) ) ).

% nth_via_drop
tff(fact_6305_drop__Nil,axiom,
    ! [A: $tType,Nb: nat] : drop(A,Nb,nil(A)) = nil(A) ).

% drop_Nil
tff(fact_6306_drop__take,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] : drop(A,Nb,take(A,Mb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Mb),Nb),drop(A,Nb,Xs)) ).

% drop_take
tff(fact_6307_take__drop,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] : take(A,Nb,drop(A,Mb,Xs)) = drop(A,Mb,take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb),Xs)) ).

% take_drop
tff(fact_6308_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_6309_distinct__drop,axiom,
    ! [A: $tType,Xs: list(A),I: nat] :
      ( distinct(A,Xs)
     => distinct(A,drop(A,I,Xs)) ) ).

% distinct_drop
tff(fact_6310_drop__eq__nths,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,Nb,Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),Nb))) ).

% drop_eq_nths
tff(fact_6311_set__drop__subset__set__drop,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Mb)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,Mb,Xs))),aa(list(A),set(A),set2(A),drop(A,Nb,Xs))) ) ).

% set_drop_subset_set_drop
tff(fact_6312_append__eq__conv__conj,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Zs )
    <=> ( ( Xs = take(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) )
        & ( Ys = drop(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) ) ) ) ).

% append_eq_conv_conj
tff(fact_6313_take__add,axiom,
    ! [A: $tType,I: nat,J: nat,Xs: list(A)] : take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I,Xs)),take(A,J,drop(A,I,Xs))) ).

% take_add
tff(fact_6314_drop__update__swap,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( drop(A,Mb,list_update(A,Xs,Nb,Xb)) = list_update(A,drop(A,Mb,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb),Xb) ) ) ).

% drop_update_swap
tff(fact_6315_drop__Cons,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : drop(A,Nb,aa(list(A),list(A),cons(A,Xb),Xs)) = case_nat(list(A),aa(list(A),list(A),cons(A,Xb),Xs),aTP_Lamp_qi(list(A),fun(nat,list(A)),Xs),Nb) ).

% drop_Cons
tff(fact_6316_nths__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),I6: set(nat)] : nths(A,drop(A,Nb,Xs),I6) = nths(A,Xs,image(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),I6)) ).

% nths_drop
tff(fact_6317_drop__Cons_H,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      drop(A,Nb,aa(list(A),list(A),cons(A,Xb),Xs)) = $ite(Nb = zero_zero(nat),aa(list(A),list(A),cons(A,Xb),Xs),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xs)) ).

% drop_Cons'
tff(fact_6318_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),Xs_2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys_1),Ys_2) )
    <=> $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)),
          ( ( Xs_1 = take(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1) )
          & ( Xs_2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1)),Ys_2) ) ),
          ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
          & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1)),Xs_2) = Ys_2 ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_6319_zip__append1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(A),Zs: list(B)] : zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),Zs) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),zip(A,B,Xs,take(B,aa(list(A),nat,size_size(list(A)),Xs),Zs))),zip(A,B,Ys,drop(B,aa(list(A),nat,size_size(list(A)),Xs),Zs))) ).

% zip_append1
tff(fact_6320_zip__append2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),zip(A,B,take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Ys)),zip(A,B,drop(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Zs)) ).

% zip_append2
tff(fact_6321_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),cons(A,aa(nat,A,nth(A,Xs),I)),drop(A,aa(nat,nat,suc,I),Xs)) = drop(A,I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_6322_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I: nat,J: nat] :
      ( distinct(A,Vs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I,Vs))),aa(list(A),set(A),set2(A),drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_6323_id__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I,Xs)),aa(list(A),list(A),cons(A,aa(nat,A,nth(A,Xs),I)),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_6324_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A),A2: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,Xs,I,A2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I,Xs)),aa(list(A),list(A),cons(A,A2),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_6325_lexn__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Nb: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),Nb) = aa(fun(product_prod(list(A),list(A)),$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_qj(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R2),Nb))) ).

% lexn_conv
tff(fact_6326_mlex__eq,axiom,
    ! [A: $tType,F3: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F3,R) = aa(fun(product_prod(A,A),$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)),aTP_Lamp_qk(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F3),R))) ).

% mlex_eq
tff(fact_6327_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R2),zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_6328_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Nb: nat] :
      ( 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,R2),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_6329_mlex__leq,axiom,
    ! [A: $tType,F3: fun(A,nat),Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,Xb)),aa(A,nat,F3,Y))
     => ( 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)
       => 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,F3,R)) ) ) ).

% mlex_leq
tff(fact_6330_mlex__iff,axiom,
    ! [A: $tType,Xb: A,Y: A,F3: fun(A,nat),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),mlex_prod(A,F3,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Xb)),aa(A,nat,F3,Y))
        | ( ( aa(A,nat,F3,Xb) = aa(A,nat,F3,Y) )
          & 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) ) ) ) ).

% mlex_iff
tff(fact_6331_mlex__less,axiom,
    ! [A: $tType,F3: fun(A,nat),Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Xb)),aa(A,nat,F3,Y))
     => 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,F3,R)) ) ).

% mlex_less
tff(fact_6332_in__measure,axiom,
    ! [A: $tType,Xb: A,Y: A,F3: fun(A,nat)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),measure(A,F3))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Xb)),aa(A,nat,F3,Y)) ) ).

% in_measure
tff(fact_6333_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F3: fun(nat,set(A)),S3: set(A)] :
      ( ! [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F3,I2)),S3)
     => ( aa(set(A),$o,finite_finite(A),S3)
       => ( ? [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),F3,M)),aa(nat,set(A),F3,N)) ) ) )
              & ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                 => ( aa(nat,set(A),F3,N6) = aa(nat,set(A),F3,N) ) ) )
         => ( aa(nat,set(A),F3,aa(set(A),nat,finite_card(A),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),F3,top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_6334_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite(option(A)),top_top(set(option(A))))
    <=> aa(set(A),$o,finite_finite(A),top_top(set(A))) ) ).

% finite_option_UNIV
tff(fact_6335_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: 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_6336_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: 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_6337_range__diff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: 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_6338_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A5) = top_top(A) )
        <=> ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),top_top(A))
             => ? [Xa2: A] :
                  ( member(A,Xa2,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xa2) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_6339_Diff__UNIV,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_6340_surj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),Nb: nat] :
      ( ( image(A,A,F3,top_top(set(A))) = top_top(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),F3),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_6341_Gcd__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).

% Gcd_UNIV
tff(fact_6342_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : image(A,A,aTP_Lamp_of(A,fun(A,A),A2),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_6343_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),image(B,A,F3,A5)) = top_top(A) )
        <=> ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),top_top(A))
             => ? [Xa2: B] :
                  ( member(B,Xa2,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),aa(B,A,F3,Xa2)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_6344_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),set_ord_lessThan(A,Xb)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_6345_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),B5: set(A),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_pr(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5),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)),image(B,set(A),A5,C5))),B5)) ).

% INT_simps(3)
tff(fact_6346_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_px(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5),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)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),B5,C5)))) ).

% INT_simps(4)
tff(fact_6347_sums__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] : sums(A,F3,aa(set(A),A,complete_Sup_Sup(A),image(nat,A,aTP_Lamp_ql(fun(nat,A),fun(nat,A),F3),top_top(set(nat))))) ) ).

% sums_SUP
tff(fact_6348_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A5) ).

% Compl_eq_Diff_UNIV
tff(fact_6349_bij__fn,axiom,
    ! [A: $tType,F3: fun(A,A),Nb: nat] :
      ( bij_betw(A,A,F3,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),F3),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_6350_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite(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)) )
       => aa(set(A),$o,finite_finite(A),top_top(set(A))) ) ) ).

% finite_fun_UNIVD1
tff(fact_6351_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_op(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_6352_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite_folding_on(A,list(A),top_top(set(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A))) ) ).

% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_6353_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_6354_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_6355_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_6356_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_6357_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_6358_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),image(A,option(A),some(A),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_6359_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_qm(fun(B,A),fun(A,$o),F3)) = image(B,A,F3,top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_6360_lex__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = 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)))),image(nat,set(product_prod(list(A),list(A))),lexn(A,R2),top_top(set(nat)))) ).

% lex_def
tff(fact_6361_finite__range__Some,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite(option(A)),image(A,option(A),some(A),top_top(set(A))))
    <=> aa(set(A),$o,finite_finite(A),top_top(set(A))) ) ).

% finite_range_Some
tff(fact_6362_notin__range__Some,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ~ member(option(A),Xb,image(A,option(A),some(A),top_top(set(A))))
    <=> ( Xb = none(A) ) ) ).

% notin_range_Some
tff(fact_6363_Collect__ex__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o))] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_qn(fun(A,fun(B,$o)),fun(A,$o),P)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),aTP_Lamp_qp(fun(A,fun(B,$o)),fun(B,set(A)),P),top_top(set(B)))) ).

% Collect_ex_eq
tff(fact_6364_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_6365_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( aa(set(A),$o,finite_finite(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_6366_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),C5: set(B),B5: 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)),image(B,set(A),A5,C5))),B5) = $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))),B5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_pr(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5),C5))) ).

% INT_extend_simps(3)
tff(fact_6367_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] :
      ( aa(set(A),$o,finite_finite(A),image(B,A,F3,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),image(B,A,F3,top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_6368_UN__finite2__eq,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B5: fun(nat,set(A)),K2: nat] :
      ( ! [N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A5,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A5,top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B5,top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_6369_suminf__eq__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] : suminf(A,F3) = aa(set(A),A,complete_Sup_Sup(A),image(nat,A,aTP_Lamp_ql(fun(nat,A),fun(nat,A),F3),top_top(set(nat)))) ) ).

% suminf_eq_SUP
tff(fact_6370_range__mod,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( image(nat,nat,aTP_Lamp_qq(nat,fun(nat,nat),Nb),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ) ).

% range_mod
tff(fact_6371_UN__finite2__subset,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B5: fun(nat,set(A)),K2: nat] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A5,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K2)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A5,top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B5,top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_6372_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),image(nat,nat,suc,top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_6373_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X3: product_prod(A,B)] :
            ( member(product_prod(A,B),X3,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
           => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X3) ) ) ) ).

% listrel_iff_zip
tff(fact_6374_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_6375_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,$o),aTP_Lamp_qr($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_6376_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_6377_range__mult,axiom,
    ! [A2: 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_6378_infinite__UNIV__listI,axiom,
    ! [A: $tType] : ~ aa(set(list(A)),$o,finite_finite(list(A)),top_top(set(list(A)))) ).

% infinite_UNIV_listI
tff(fact_6379_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_6380_listrel__Nil2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),nil(B)),listrel(A,B,R2))
     => ( Xs = nil(A) ) ) ).

% listrel_Nil2
tff(fact_6381_listrel__Nil1,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xs),listrel(A,B,R2))
     => ( Xs = nil(B) ) ) ).

% listrel_Nil1
tff(fact_6382_listrel_ONil,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)),listrel(A,B,R2)) ).

% listrel.Nil
tff(fact_6383_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel_eq_len
tff(fact_6384_surj__list__encode,axiom,
    image(list(nat),nat,nat_list_encode,top_top(set(list(nat)))) = top_top(set(nat)) ).

% surj_list_encode
tff(fact_6385_bij__list__encode,axiom,
    bij_betw(list(nat),nat,nat_list_encode,top_top(set(list(nat))),top_top(set(nat))) ).

% bij_list_encode
tff(fact_6386_surj__prod__encode,axiom,
    image(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat)))) = top_top(set(nat)) ).

% surj_prod_encode
tff(fact_6387_bij__prod__encode,axiom,
    bij_betw(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat))),top_top(set(nat))) ).

% bij_prod_encode
tff(fact_6388_listrel__mono,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S2)
     => aa(set(product_prod(list(A),list(B))),$o,aa(set(product_prod(list(A),list(B))),fun(set(product_prod(list(A),list(B))),$o),ord_less_eq(set(product_prod(list(A),list(B)))),listrel(A,B,R2)),listrel(A,B,S2)) ) ).

% listrel_mono
tff(fact_6389_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,R2: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y),R2)
     => ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
       => member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(B),list(B),cons(B,Y),Ys)),listrel(A,B,R2)) ) ) ).

% listrel.Cons
tff(fact_6390_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),cons(A,Y),Ys)),Xs),listrel(A,B,R2))
     => ~ ! [Y3: B,Ys3: list(B)] :
            ( ( Xs = aa(list(B),list(B),cons(B,Y3),Ys3) )
           => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y3),R2)
             => ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Ys),Ys3),listrel(A,B,R2)) ) ) ) ).

% listrel_Cons1
tff(fact_6391_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),cons(B,Y),Ys)),listrel(A,B,R2))
     => ~ ! [X4: A,Xs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),cons(A,X4),Xs2) )
           => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y),R2)
             => ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys),listrel(A,B,R2)) ) ) ) ).

% listrel_Cons2
tff(fact_6392_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X4: A,Y3: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),cons(A,X4),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),cons(B,Y3),Ys3) )
                 => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R2)
                   => ~ member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys3),listrel(A,B,R2)) ) ) ) ) ) ).

% listrel.cases
tff(fact_6393_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R2))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X3: A,Y5: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),cons(A,X3),Xs3) )
            & ( A22 = aa(list(B),list(B),cons(B,Y5),Ys4) )
            & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y5),R2)
            & member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys4),listrel(A,B,R2)) ) ) ) ).

% listrel.simps
tff(fact_6394_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N4),aa(list(A),nat,size_size(list(A)),Xs))
           => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N4)),aa(nat,B,nth(B,Ys),N4)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_6395_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_qs(nat,fun(real,real),Nb),Xb)) ).

% root_def
tff(fact_6396_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_6397_UNIV__char__of__nat,axiom,
    top_top(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_6398_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_6399_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_6400_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_6401_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_6402_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A] : aa(A,char,unique5772411509450598832har_of(A),Nb) = char2(~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),Nb),aa(nat,$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_6403_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [C3: char] : modulo_modulo(A,aa(char,A,comm_s6883823935334413003f_char(A),C3),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),C3) ) ).

% of_char_mod_256
tff(fact_6404_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: $o] : aa(char,A,comm_s6883823935334413003f_char(A),char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(list($o),list($o),cons($o,(B0)),aa(list($o),list($o),cons($o,(B1)),aa(list($o),list($o),cons($o,(B22)),aa(list($o),list($o),cons($o,(B32)),aa(list($o),list($o),cons($o,(B42)),aa(list($o),list($o),cons($o,(B52)),aa(list($o),list($o),cons($o,(B62)),aa(list($o),list($o),cons($o,(B72)),nil($o)))))))))) ) ).

% of_char_Char
tff(fact_6405_char_Osize_I2_J,axiom,
    ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : aa(char,nat,size_size(char),char2((X1),(X2),(X32),(X42),(X52),(X62),(X72),(X8))) = zero_zero(nat) ).

% char.size(2)
tff(fact_6406_nat__of__char__less__256,axiom,
    ! [C3: char] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C3)),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_6407_range__nat__of__char,axiom,
    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_6408_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A,C3: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),Nb) = C3 )
        <=> ( 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),C3) ) ) ) ).

% char_of_eq_iff
tff(fact_6409_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa($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_6410_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),Js,upto_aux(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),aa(list(int),list(int),cons(int,J),Js))) ).

% upto_aux_rec
tff(fact_6411_String_Ochar__of__ascii__of,axiom,
    ! [C3: char] : aa(char,nat,comm_s6883823935334413003f_char(nat),ascii_of(C3)) = 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),C3)) ).

% String.char_of_ascii_of
tff(fact_6412_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_6413_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil(int) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil
tff(fact_6414_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil(int) = upto(I,J) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil2
tff(fact_6415_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I)
     => ( upto(I,J) = nil(int) ) ) ).

% upto_empty
tff(fact_6416_upto__single,axiom,
    ! [I: int] : upto(I,I) = aa(list(int),list(int),cons(int,I),nil(int)) ).

% upto_single
tff(fact_6417_nth__upto,axiom,
    ! [I: int,K2: 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),I),aa(nat,int,semiring_1_of_nat(int),K2))),J)
     => ( aa(nat,int,nth(int,upto(I,J)),K2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K2)) ) ) ).

% nth_upto
tff(fact_6418_length__upto,axiom,
    ! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,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),I)),one_one(int))) ).

% length_upto
tff(fact_6419_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)),aa(list(int),list(int),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_6420_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))),aa(list(int),list(int),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_6421_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)),aa(list(int),list(int),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_6422_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))),aa(list(int),list(int),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_6423_upto__code,axiom,
    ! [I: int,J: int] : upto(I,J) = upto_aux(I,J,nil(int)) ).

% upto_code
tff(fact_6424_upto__aux__def,axiom,
    ! [I: int,J: int,Js: list(int)] : upto_aux(I,J,Js) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),Js) ).

% upto_aux_def
tff(fact_6425_distinct__upto,axiom,
    ! [I: int,J: int] : distinct(int,upto(I,J)) ).

% distinct_upto
tff(fact_6426_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1337092689740270186AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_6427_upto__split2,axiom,
    ! [I: int,J: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K2)
       => ( upto(I,K2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K2)) ) ) ) ).

% upto_split2
tff(fact_6428_upto__split1,axiom,
    ! [I: int,J: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K2)
       => ( upto(I,K2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),upto(J,K2)) ) ) ) ).

% upto_split1
tff(fact_6429_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6430_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_6431_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),aa(list(int),list(int),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_6432_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_6433_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_6434_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),cons(int,J),nil(int))) ) ) ).

% upto_rec2
tff(fact_6435_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_6436_upto__split3,axiom,
    ! [I: int,J: int,K2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K2)
       => ( upto(I,K2) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),cons(int,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K2))) ) ) ) ).

% upto_split3
tff(fact_6437_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),aa(list(int),list(int),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_6438_take__hd__drop,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(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Nb,Xs)),aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),drop(A,Nb,Xs))),nil(A))) = take(A,aa(nat,nat,suc,Nb),Xs) ) ) ).

% take_hd_drop
tff(fact_6439_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs: 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),Zs))) )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),cons(A,Y),Zs)) )
        & aa(A,$o,P,Y)
        & ~ ? [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X) ) ) ) ).

% extract_SomeE
tff(fact_6440_hd__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_append2
tff(fact_6441_hd__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),A,hd(A),replicate(A,Nb,Xb)) = Xb ) ) ).

% hd_replicate
tff(fact_6442_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),J)
     => ( aa(list(A),A,hd(A),take(A,J,Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_take
tff(fact_6443_hd__in__set,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => member(A,aa(list(A),A,hd(A),Xs),aa(list(A),set(A),set2(A),Xs)) ) ).

% hd_in_set
tff(fact_6444_list_Oset__sel_I1_J,axiom,
    ! [A: $tType,A2: list(A)] :
      ( ( A2 != nil(A) )
     => member(A,aa(list(A),A,hd(A),A2),aa(list(A),set(A),set2(A),A2)) ) ).

% list.set_sel(1)
tff(fact_6445_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( aa(list(product_prod(A,B)),product_prod(A,B),hd(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(list(A),A,hd(A),Xs)),aa(list(B),B,hd(B),Ys)) ) ) ) ).

% hd_zip
tff(fact_6446_hd__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( Xs != nil(list(A)) )
     => ( ( aa(list(list(A)),list(A),hd(list(A)),Xs) != nil(A) )
       => ( aa(list(A),A,hd(A),concat(A,Xs)) = aa(list(A),A,hd(A),aa(list(list(A)),list(A),hd(list(A)),Xs)) ) ) ) ).

% hd_concat
tff(fact_6447_extract__Nil__code,axiom,
    ! [A: $tType,P: fun(A,$o)] : extract(A,P,nil(A)) = none(product_prod(list(A),product_prod(A,list(A)))) ).

% extract_Nil_code
tff(fact_6448_hd__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(Xs = nil(A),aa(list(A),A,hd(A),Ys),aa(list(A),A,hd(A),Xs)) ).

% hd_append
tff(fact_6449_longest__common__prefix,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
    ? [Ps2: list(A),Xs4: list(A),Ys5: list(A)] :
      ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Xs4) )
      & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps2),Ys5) )
      & ( ( Xs4 = nil(A) )
        | ( Ys5 = nil(A) )
        | ( aa(list(A),A,hd(A),Xs4) != aa(list(A),A,hd(A),Ys5) ) ) ) ).

% longest_common_prefix
tff(fact_6450_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)))) )
    <=> ~ ? [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X3) ) ) ).

% extract_None_iff
tff(fact_6451_list_Osel_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),A,hd(A),aa(list(A),list(A),cons(A,X21),X222)) = X21 ).

% list.sel(1)
tff(fact_6452_hd__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_6453_hd__drop__conv__nth,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(list(A),A,hd(A),drop(A,Nb,Xs)) = aa(nat,A,nth(A,Xs),Nb) ) ) ).

% hd_drop_conv_nth
tff(fact_6454_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      extract(A,P,aa(list(A),list(A),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_qu(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_6455_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs: 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),Zs))) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),cons(A,Y),Zs)) )
        & aa(A,$o,P,Y)
        & ~ ? [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X3) ) ) ) ).

% extract_Some_iff
tff(fact_6456_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D4: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( ( Xb != zero_zero(real) )
       => ( ( 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)
             => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),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)))))) ) ) )
         => ( ( 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))
               => ( D4 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),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))))))) ) ) )
           => ( ( ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Nb)
               => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),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),D4,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_6457_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( 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_6458_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Y: A,Z: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F3,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xb),image(A,A,aa(A,fun(A,A),plus_plus(A),Z),S3)))
         => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_6459_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,G3: fun(A,A),Xb: A,S2: set(A),Db: A] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,aa(A,A,G3,Xb),image(A,A,G3,S2)))
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xb,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),G3),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_image_chain
tff(fact_6460_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Y: A,Z: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F3,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xb),image(A,A,aa(A,fun(A,A),plus_plus(A),Z),S3)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_qv(fun(A,A),fun(A,fun(A,A)),F3),Z),Y,topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_at_within_shift
tff(fact_6461_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => ( ( aa(A,A,F3,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_qw(fun(A,A),fun(A,A),F3),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,F3,Xb))),D4)),aa(A,A,inverse_inverse(A),aa(A,A,F3,Xb)))),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_inverse'
tff(fact_6462_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,Xb,S2))
           => ( ( aa(A,A,G3,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qx(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),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),D4),aa(A,A,G3,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F3,Xb)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G3,Xb)),aa(A,A,G3,Xb))),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% DERIV_divide
tff(fact_6463_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),C3: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_qy(fun(A,A),fun(A,fun(A,A)),F3),C3),divide_divide(A,D4,C3),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_cdivide
tff(fact_6464_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),C3: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_qz(fun(A,A),fun(A,fun(A,A)),F3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D4),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_cmult
tff(fact_6465_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),C3: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_ra(fun(A,A),fun(A,fun(A,A)),F3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),D4),C3),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_cmult_right
tff(fact_6466_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,Xb: A,S2: set(A),G3: fun(A,A),Db: A] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,Xb,S2))
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xb,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rb(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),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,G3,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F3,Xb))),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_mult
tff(fact_6467_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,Xb,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rb(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),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,F3,Xb)),E5)),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(A,A,G3,Xb))),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_mult'
tff(fact_6468_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,Xb,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rc(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),D4),E5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_add
tff(fact_6469_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),F6: A,F4: filter(A),G3: fun(A,A),G5: A] :
          ( has_field_derivative(A,F3,F6,F4)
         => ( has_field_derivative(A,G3,G5,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rc(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),plus_plus(A),F6),G5),F4) ) ) ) ).

% field_differentiable_add
tff(fact_6470_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F4: filter(A)] : has_field_derivative(A,aTP_Lamp_rd(A,A),one_one(A),F4) ) ).

% DERIV_ident
tff(fact_6471_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),F6: A,F4: filter(A),G3: fun(A,A),G5: A] :
          ( has_field_derivative(A,F3,F6,F4)
         => ( has_field_derivative(A,G3,G5,F4)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_re(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),minus_minus(A),F6),G5),F4) ) ) ) ).

% field_differentiable_diff
tff(fact_6472_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,Xb,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_re(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),minus_minus(A),D4),E5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_diff
tff(fact_6473_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Db: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xb,S2))
         => has_field_derivative(A,aTP_Lamp_rf(fun(A,A),fun(A,A),G3),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G3,Xb))),Db),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_field_derivative_sinh
tff(fact_6474_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Db: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xb,S2))
         => has_field_derivative(A,aTP_Lamp_rg(fun(A,A),fun(A,A),G3),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G3,Xb))),Db),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_field_derivative_cosh
tff(fact_6475_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,Xb: A,S2: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C3),C3,topolo174197925503356063within(A,Xb,S2)) ) ).

% DERIV_cmult_Id
tff(fact_6476_has__real__derivative__pos__inc__left,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4),S3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))),aa(real,real,F3,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_6477_has__real__derivative__neg__dec__left,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [H4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H4)
               => ( member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4),S3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H4),D2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xb)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_6478_MVT2,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),F6: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [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)
             => has_field_derivative(real,F3,aa(real,real,F6,X4),topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
       => ? [Z3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
            & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,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,Z3)) ) ) ) ) ).

% MVT2
tff(fact_6479_DERIV__neg__dec__left,axiom,
    ! [F3: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [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),D2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xb)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_6480_DERIV__pos__inc__left,axiom,
    ! [F3: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
            & ! [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),D2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xb),H4))),aa(real,real,F3,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_6481_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),K2: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : has_field_derivative(real,F3,K2,topolo174197925503356063within(real,X4,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,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)),K2) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_6482_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Y: A,Xb: A,Z: A] :
          ( has_field_derivative(A,F3,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_rh(fun(A,A),fun(A,fun(A,A)),F3),Z),Y,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_6483_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Mb: A,Xb: A] :
          ( has_field_derivative(A,G3,Mb,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_ri(fun(A,A),fun(A,A),G3),aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,aa(A,A,G3,Xb))),Mb),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_6484_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Mb: A,Xb: A] :
          ( has_field_derivative(A,G3,Mb,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_rj(fun(A,A),fun(A,A),G3),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G3,Xb))),Mb),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_6485_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),G3: fun(A,A),E5: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => ( has_field_derivative(A,G3,E5,topolo174197925503356063within(A,aa(A,A,F3,Xb),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rk(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),times_times(A),E5),D4),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_chain'
tff(fact_6486_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,G3: fun(A,A),Xb: A,Db: A,S2: set(A)] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,aa(A,A,G3,Xb),top_top(set(A))))
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xb,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rl(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_chain2
tff(fact_6487_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G3: fun(A,A),G5: fun(A,A),F3: fun(A,A),F6: A,Xb: A] :
          ( ! [X4: A] : has_field_derivative(A,G3,aa(A,A,G5,X4),topolo174197925503356063within(A,X4,top_top(set(A))))
         => ( has_field_derivative(A,F3,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rl(fun(A,A),fun(fun(A,A),fun(A,A)),G3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G5,aa(A,A,F3,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_6488_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S2: set(A),G3: fun(A,A),G5: fun(A,A),F3: fun(A,A),F6: A,Xb: A] :
          ( ! [X4: A] :
              ( member(A,X4,S2)
             => has_field_derivative(A,G3,aa(A,A,G5,X4),topolo174197925503356063within(A,X4,top_top(set(A)))) )
         => ( has_field_derivative(A,F3,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => ( member(A,aa(A,A,F3,Xb),S2)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_rl(fun(A,A),fun(fun(A,A),fun(A,A)),G3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),F6),aa(A,A,G5,aa(A,A,F3,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_6489_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Mb: A,Xb: A] :
          ( has_field_derivative(A,G3,Mb,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_rm(fun(A,A),fun(A,A),G3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,G3,Xb)))),Mb),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_6490_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K2: A,Xaa: A] : has_field_derivative(A,aTP_Lamp_rn(A,fun(A,A),K2),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xaa),K2))),topolo174197925503356063within(A,Xaa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_6491_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),K2: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : has_field_derivative(real,F3,K2,topolo174197925503356063within(real,X4,top_top(set(real))))
       => ( divide_divide(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,A2)),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)) = K2 ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_6492_DERIV__local__const,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,D3: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
       => ( ! [Y3: 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),Y3))),D3)
             => ( aa(real,real,F3,Xb) = aa(real,real,F3,Y3) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_6493_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,G3: fun(A,A),Xb: A,Db: A,S2: set(A)] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,aa(A,A,G3,Xb),top_top(set(A))))
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xb,S2))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),G3),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_chain
tff(fact_6494_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),Nb: nat] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_ro(fun(A,A),fun(nat,fun(A,A)),F3),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),D4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F3,Xb)),Nb))),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_power_Suc
tff(fact_6495_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun(real,real),K2: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : has_field_derivative(real,V,K2,topolo174197925503356063within(real,X4,top_top(set(real))))
       => ( aa(real,real,V,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)))) = 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_6496_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,S2: 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,S2)) ) ) ).

% DERIV_inverse
tff(fact_6497_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S2: set(A),Nb: nat] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S2))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_rp(fun(A,A),fun(nat,fun(A,A)),F3),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),D4),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F3,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_power
tff(fact_6498_DERIV__local__min,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,D3: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
       => ( ! [Y3: 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),Y3))),D3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F3,Xb)),aa(real,real,F3,Y3)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_6499_DERIV__local__max,axiom,
    ! [F3: fun(real,real),L: real,Xb: real,D3: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
       => ( ! [Y3: 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),Y3))),D3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F3,Y3)),aa(real,real,F3,Xb)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_6500_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),divide_divide(real,one_one(real),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_6501_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S2: set(real)] : has_field_derivative(real,aTP_Lamp_rq(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,S2)) ).

% DERIV_pow
tff(fact_6502_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),Xb: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C3),Y3))
         => has_field_derivative(A,aTP_Lamp_rr(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C3),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_6503_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_6504_DERIV__fun__pow,axiom,
    ! [G3: fun(real,real),Mb: real,Xb: real,Nb: nat] :
      ( has_field_derivative(real,G3,Mb,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_rs(fun(real,real),fun(nat,fun(real,real)),G3),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,G3,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_6505_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,set_ord_lessThan(A,B2)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_6506_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D3: A,Xb: A,S2: set(A),G3: fun(A,A),E3: A] :
          ( has_field_derivative(A,F3,D3,topolo174197925503356063within(A,Xb,S2))
         => ( has_field_derivative(A,G3,E3,topolo174197925503356063within(A,Xb,S2))
           => ( ( aa(A,A,G3,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_qx(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),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),D3),aa(A,A,G3,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),E3),aa(A,A,F3,Xb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G3,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% DERIV_quotient
tff(fact_6507_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D3: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,F3,D3,topolo174197925503356063within(A,Xb,S2))
         => ( ( aa(A,A,F3,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_qw(fun(A,A),fun(A,A),F3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F3,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_6508_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C3: fun(nat,A),F3: fun(A,A),F6: A,Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K5)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C3),Z3),aa(A,A,F3,Z3)) )
         => ( has_field_derivative(A,F3,F6,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C3),Z),F6) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_6509_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_rt(real,fun(real,real),R2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,Z,aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_6510_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ru(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
             => ( aa(real,$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_rr(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C3),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_6511_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( aa(real,$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_rr(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C3),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_6512_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C3: fun(nat,A),Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K5)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C3),Z3)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5)
           => has_field_derivative(A,aTP_Lamp_rr(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),C3),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_6513_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),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_6514_DERIV__fun__powr,axiom,
    ! [G3: fun(real,real),Mb: real,Xb: real,R2: real] :
      ( has_field_derivative(real,G3,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,G3,Xb))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_rv(fun(real,real),fun(real,fun(real,real)),G3),R2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,aa(real,real,G3,Xb),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),Mb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_6515_DERIV__powr,axiom,
    ! [G3: fun(real,real),Mb: real,Xb: real,F3: fun(real,real),R2: real] :
      ( has_field_derivative(real,G3,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,G3,Xb))
       => ( has_field_derivative(real,F3,R2,topolo174197925503356063within(real,Xb,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_rw(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F3),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G3,Xb),aa(real,real,F3,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),aa(real,real,ln_ln(real),aa(real,real,G3,Xb)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),Mb),aa(real,real,F3,Xb)),aa(real,real,G3,Xb)))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_6516_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_6517_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,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_6518_DERIV__series_H,axiom,
    ! [F3: 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_rx(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F3),N),aa(nat,real,aa(real,fun(nat,real),F6,X0),N),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X4: real] :
            ( member(real,X4,set_or5935395276787703475ssThan(real,A2,B2))
           => summable(real,aa(real,fun(nat,real),F3,X4)) )
       => ( member(real,X0,set_or5935395276787703475ssThan(real,A2,B2))
         => ( summable(real,aa(real,fun(nat,real),F6,X0))
           => ( summable(real,L5)
             => ( ! [N: nat,X4: real,Y3: real] :
                    ( member(real,X4,set_or5935395276787703475ssThan(real,A2,B2))
                   => ( member(real,Y3,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),F3,X4),N)),aa(nat,real,aa(real,fun(nat,real),F3,Y3),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),X4),Y3)))) ) )
               => has_field_derivative(real,aTP_Lamp_ry(fun(real,fun(nat,real)),fun(real,real),F3),suminf(real,aa(real,fun(nat,real),F6,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_6519_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_6520_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A5: set(real)] : has_field_derivative(real,arsinh(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,A5)) ).

% arsinh_real_has_field_derivative
tff(fact_6521_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_6522_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Xb: A,Db: A,S2: set(A)] :
          ( ( cosh(A,aa(A,A,G3,Xb)) != zero_zero(A) )
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xb,S2))
           => has_field_derivative(A,aTP_Lamp_rz(fun(A,A),fun(A,A),G3),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,G3,Xb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Db),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_6523_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D4: real] :
      ( ( Xb != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xb)
         => ( D4 = 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))
           => ( D4 = 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,D4,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_6524_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A5: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xb)
     => has_field_derivative(real,arcosh(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,A5)) ) ).

% arcosh_real_has_field_derivative
tff(fact_6525_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A5: 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),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,A5)) ) ).

% artanh_real_has_field_derivative
tff(fact_6526_DERIV__power__series_H,axiom,
    ! [R: real,F3: fun(nat,real),X0: real] :
      ( ! [X4: real] :
          ( member(real,X4,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_sa(fun(nat,real),fun(real,fun(nat,real)),F3),X4)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => has_field_derivative(real,aTP_Lamp_sc(fun(nat,real),fun(real,real),F3),suminf(real,aa(real,fun(nat,real),aTP_Lamp_sa(fun(nat,real),fun(real,fun(nat,real)),F3),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_6527_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_6528_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_6529_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_6530_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Xb: real,Nb: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
        & ! [M: nat,X4: 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)),X4),topolo174197925503356063within(real,X4,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,F3,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_sd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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_6531_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Xb: real,Nb: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [M: nat,X4: 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)),X4),topolo174197925503356063within(real,X4,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,F3,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_sd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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_6532_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ 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_6533_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F3: 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)) = F3 )
         => ( ! [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,F3,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_se(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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_6534_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F3: 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)) = F3 )
       => ( ! [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,F3,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_se(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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_6535_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun(nat,fun(real,real)),F3: 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)) = F3 )
         => ( ! [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,F3,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_se(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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_6536_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => ( ( Xb != zero_zero(real) )
         => ( ! [M: nat,X4: 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)),X4),topolo174197925503356063within(real,X4,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,F3,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_sd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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_6537_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Nb: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [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,F3,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_sd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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_6538_Taylor,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C3: 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)) = F3 )
       => ( ! [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),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),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 != C3 )
                   => ? [T4: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),C3),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),T4)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),C3) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),T4)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),Xb) ) )
                        & ( aa(real,real,F3,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_sf(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C3),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),C3)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_6539_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C3: 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)) = F3 )
       => ( ! [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),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),B2)
             => ? [T4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),T4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T4),B2)
                  & ( aa(real,real,F3,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_sg(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C3)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),C3)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_6540_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C3: 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)) = F3 )
       => ( ! [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),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),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),C3)
                  & ( aa(real,real,F3,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_sg(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C3)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(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),C3)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_6541_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun(nat,fun(real,real)),K2: nat,B5: 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,K2) )
       => ! [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_si(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Nb),Diff),B5),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_sj(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T6)),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),B5),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_6542_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_sk(real,real),suminf(real,aTP_Lamp_sl(real,fun(nat,real),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_6543_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xb: A,G5: fun(A,real),S2: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G3,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G3,Xb)),one_one(real))
           => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
             => has_derivative(A,real,aTP_Lamp_sm(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sn(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xb),G5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_6544_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xb: A,G5: fun(A,real),S2: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G3,Xb))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G3,Xb)),one_one(real))
           => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
             => has_derivative(A,real,aTP_Lamp_so(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_sp(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xb),G5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_6545_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,real),F6: fun(A,real),Xb: A,S2: set(A),G3: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,real,F3,F6,topolo174197925503356063within(A,Xb,S2))
         => ( has_derivative(A,B,G3,G5,topolo174197925503356063within(A,Xb,S2))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sq(fun(A,real),fun(fun(A,B),fun(A,B)),F3),G3),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_sr(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F6),Xb),G3),G5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_derivative_scaleR
tff(fact_6546_has__field__derivative__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,F4: filter(A)] :
          ( has_field_derivative(A,F3,D4,F4)
        <=> has_derivative(A,A,F3,aa(A,fun(A,A),times_times(A),D4),F4) ) ) ).

% has_field_derivative_def
tff(fact_6547_has__derivative__imp__has__field__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: fun(A,A),F4: filter(A),D7: A] :
          ( has_derivative(A,A,F3,D4,F4)
         => ( ! [X4: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X4),D7) = aa(A,A,D4,X4)
           => has_field_derivative(A,F3,D7,F4) ) ) ) ).

% has_derivative_imp_has_field_derivative
tff(fact_6548_has__field__derivative__imp__has__derivative,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,F4: filter(A)] :
          ( has_field_derivative(A,F3,D4,F4)
         => has_derivative(A,A,F3,aa(A,fun(A,A),times_times(A),D4),F4) ) ) ).

% has_field_derivative_imp_has_derivative
tff(fact_6549_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),F4: filter(A),G3: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F3,F6,F4)
         => ( has_derivative(A,B,G3,G5,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ss(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),aa(fun(A,B),fun(A,B),aTP_Lamp_ss(fun(A,B),fun(fun(A,B),fun(A,B)),F6),G5),F4) ) ) ) ).

% has_derivative_diff
tff(fact_6550_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),F4: filter(A),G3: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F3,F6,F4)
         => ( has_derivative(A,B,G3,G5,F4)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_st(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),aa(fun(A,B),fun(A,B),aTP_Lamp_st(fun(A,B),fun(fun(A,B),fun(A,B)),F6),G5),F4) ) ) ) ).

% has_derivative_add
tff(fact_6551_has__derivative__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G3: fun(A,B),G5: fun(A,B),F4: filter(A),Xb: B] :
          ( has_derivative(A,B,G3,G5,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_su(fun(A,B),fun(B,fun(A,B)),G3),Xb),aa(B,fun(A,B),aTP_Lamp_su(fun(A,B),fun(B,fun(A,B)),G5),Xb),F4) ) ) ).

% has_derivative_mult_right
tff(fact_6552_has__derivative__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G3: fun(A,B),G5: fun(A,B),F4: filter(A),Y: B] :
          ( has_derivative(A,B,G3,G5,F4)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_sv(fun(A,B),fun(B,fun(A,B)),G3),Y),aa(B,fun(A,B),aTP_Lamp_sv(fun(A,B),fun(B,fun(A,B)),G5),Y),F4) ) ) ).

% has_derivative_mult_left
tff(fact_6553_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),Xb: A,S2: set(A),G3: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S2))
         => ( has_derivative(A,B,G3,G5,topolo174197925503356063within(A,Xb,S2))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sw(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),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_sx(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F6),Xb),G3),G5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_derivative_mult
tff(fact_6554_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),Xb: A,S2: set(A)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
         => has_derivative(A,real,aTP_Lamp_sy(fun(A,real),fun(A,real),G3),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_sz(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G3),G5),Xb),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_exp
tff(fact_6555_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),Xb: A,S2: set(A)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
         => has_derivative(A,real,aTP_Lamp_ta(fun(A,real),fun(A,real),G3),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tb(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G3),G5),Xb),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_sin
tff(fact_6556_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Db: A,Xb: A,S2: set(A)] :
          ( has_derivative(A,A,G3,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xb,S2))
         => has_derivative(A,A,aTP_Lamp_rg(fun(A,A),fun(A,A),G3),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G3,Xb))),Db)),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_cosh
tff(fact_6557_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Db: A,Xb: A,S2: set(A)] :
          ( has_derivative(A,A,G3,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xb,S2))
         => has_derivative(A,A,aTP_Lamp_rf(fun(A,A),fun(A,A),G3),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G3,Xb))),Db)),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_sinh
tff(fact_6558_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),G3: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G3,G5,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,G3,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tc(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),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_td(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F6),Xb),G3),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_6559_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,S3: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_te(A,fun(A,A),Xb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_inverse'
tff(fact_6560_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(B,A),Xb: B,F6: fun(B,A),S3: set(B)] :
          ( ( aa(B,A,F3,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F3,F6,topolo174197925503356063within(B,Xb,S3))
           => has_derivative(B,A,aTP_Lamp_tf(fun(B,A),fun(B,A),F3),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_tg(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F3),Xb),F6),topolo174197925503356063within(B,Xb,S3)) ) ) ) ).

% has_derivative_inverse
tff(fact_6561_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(real,real),F6: real,G3: fun(A,real),Xb: A,G5: fun(A,real),S2: set(A)] :
          ( has_field_derivative(real,F3,F6,topolo174197925503356063within(real,aa(A,real,G3,Xb),top_top(set(real))))
         => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_th(fun(real,real),fun(fun(A,real),fun(A,real)),F3),G3),aa(fun(A,real),fun(A,real),aTP_Lamp_ti(real,fun(fun(A,real),fun(A,real)),F6),G5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_6562_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),Xb: A,S2: set(A)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
         => has_derivative(A,real,aTP_Lamp_tj(fun(A,real),fun(A,real),G3),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_tk(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G3),G5),Xb),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_cos
tff(fact_6563_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),Nb: nat] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_tl(fun(A,B),fun(nat,fun(A,B)),F3),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_tm(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F3),F6),Xb),Nb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_power
tff(fact_6564_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xb: A,G5: fun(A,real),S2: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,Xb))
         => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
           => has_derivative(A,real,aTP_Lamp_tn(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_to(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xb),G5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_derivative_ln
tff(fact_6565_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),Xb: A,S3: set(A),G3: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G3,G5,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,G3,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tp(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),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_tq(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F6),Xb),G3),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_divide
tff(fact_6566_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I6: set(A),F3: fun(A,fun(B,C)),F6: fun(A,fun(B,C)),Xb: B,S3: set(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I6)
             => has_derivative(B,C,aa(A,fun(B,C),F3,I2),aa(A,fun(B,C),F6,I2),topolo174197925503356063within(B,Xb,S3)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ts(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I6),F3),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_tu(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I6),F3),F6),Xb),topolo174197925503356063within(B,Xb,S3)) ) ) ).

% has_derivative_prod
tff(fact_6567_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),Xb: A,X6: set(A),F3: fun(A,real),F6: fun(A,real)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,X6))
         => ( has_derivative(A,real,F3,F6,topolo174197925503356063within(A,Xb,X6))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,Xb))
             => ( member(A,Xb,X6)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tv(fun(A,real),fun(fun(A,real),fun(A,real)),G3),F3),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_tw(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G3),G5),Xb),F3),F6),topolo174197925503356063within(A,Xb,X6)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_6568_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xb: A,G5: fun(A,real),S2: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,Xb))
         => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
           => has_derivative(A,real,aTP_Lamp_tx(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ty(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xb),G5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_6569_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G5: fun(A,real),Xb: A,S2: set(A)] :
          ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
         => has_derivative(A,real,aTP_Lamp_tz(fun(A,real),fun(A,real),G3),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ua(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G3),G5),Xb),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% has_derivative_arctan
tff(fact_6570_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xb: A,G5: fun(A,real),S2: set(A)] :
          ( ( cos(real,aa(A,real,G3,Xb)) != zero_zero(real) )
         => ( has_derivative(A,real,G3,G5,topolo174197925503356063within(A,Xb,S2))
           => has_derivative(A,real,aTP_Lamp_ub(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_uc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xb),G5),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_derivative_tan
tff(fact_6571_has__derivative__floor,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [G3: fun(B,real),Xb: B,F3: fun(real,A),G5: fun(B,real),S2: set(B)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,aa(B,real,G3,Xb),top_top(set(real))),F3)
         => ( ~ member(A,aa(real,A,F3,aa(B,real,G3,Xb)),ring_1_Ints(A))
           => ( has_derivative(B,real,G3,G5,topolo174197925503356063within(B,Xb,S2))
             => has_derivative(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_ud(fun(B,real),fun(fun(real,A),fun(B,real)),G3),F3),aTP_Lamp_ue(fun(B,real),fun(B,real),G5),topolo174197925503356063within(B,Xb,S2)) ) ) ) ) ).

% has_derivative_floor
tff(fact_6572_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ru(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( aa(real,$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_ug(fun(nat,A),fun(A,fun(A,A)),C3),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_6573_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(B,A),L: A,F4: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uh(A,fun(fun(B,A),fun(B,A)),C3),F3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C3)),F4)
          <=> filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_6574_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(B,A),L: A,F4: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ui(A,fun(fun(B,A),fun(B,A)),C3),F3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),L)),F4)
          <=> filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_6575_power__tendsto__0__iff,axiom,
    ! [A: $tType,Nb: nat,F3: fun(A,real),F4: filter(A)] :
      ( aa(nat,$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_uj(nat,fun(fun(A,real),fun(A,real)),Nb),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_6576_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,F3: fun(A,B),G3: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => 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_uk(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3)) ) ) ) ).

% isCont_Pair
tff(fact_6577_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S3))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ul(fun(A,A),fun(A,fun(A,A)),F3),Xb),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivativeD
tff(fact_6578_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,S3))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ul(fun(A,A),fun(A,fun(A,A)),F3),Xb),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivative_iff
tff(fact_6579_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L: B,A2: A,G3: fun(A,C),Mb: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X4: A] :
                ( ( X4 != A2 )
               => aa(real,$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,G3,X4)),Mb))),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F3,X4)),L))) )
           => filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,Mb),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_6580_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,K2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_um(fun(A,B),fun(A,fun(A,B)),F3),K2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),K2),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_6581_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_un(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_6582_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_un(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_6583_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_un(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_6584_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xb: A,F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),F3)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_uo(A,fun(fun(A,B),fun(A,B)),Xb),F3),topolo7230453075368039082e_nhds(B,aa(A,B,F3,Xb)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_6585_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F3: fun(A,B),G3: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( filterlim(B,C,G3,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F3,A2),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$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))),D6) )
                     => ( aa(A,B,F3,X4) != aa(A,B,F3,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_up(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_6586_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,F3: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( filterlim(B,C,F3,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_uq(B,fun(fun(B,C),fun(B,C)),A2),F3),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_6587_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,C),B2: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,B2),F4)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ur(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3),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_6588_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,C,F4,G3)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F4,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_uk(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3)) ) ) ) ).

% continuous_Pair
tff(fact_6589_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I6: set(A),F3: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I6)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_us(fun(B,fun(A,C)),fun(A,fun(B,C)),F3),I2),topolo7230453075368039082e_nhds(C,one_one(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_ut(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I6),F3),topolo7230453075368039082e_nhds(C,one_one(C)),F4) ) ) ).

% tendsto_one_prod'
tff(fact_6590_tendsto__artanh,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$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_uu(fun(A,real),fun(A,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_6591_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F3: fun(A,B),Nb: nat] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => topolo3448309680560233919inuous(A,B,F4,aa(nat,fun(A,B),aTP_Lamp_uv(fun(A,B),fun(nat,fun(A,B)),F3),Nb)) ) ) ).

% continuous_power
tff(fact_6592_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: fun(A,B),A2: B,F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_uw(fun(A,B),fun(nat,fun(A,B)),F3),Nb),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),Nb)),F4) ) ) ).

% tendsto_power
tff(fact_6593_continuous__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,nat)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,nat,F4,G3)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,nat),fun(A,B),aTP_Lamp_ux(fun(A,B),fun(fun(A,nat),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_power'
tff(fact_6594_tendsto__power__strong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,nat),B2: nat] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,nat,G3,topolo7230453075368039082e_nhds(nat,B2),F4)
           => filterlim(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_uy(fun(A,B),fun(fun(A,nat),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A2),B2)),F4) ) ) ) ).

% tendsto_power_strong
tff(fact_6595_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F3: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_uz(fun(A,B),fun(nat,fun(A,B)),F3),Nb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_6596_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_va(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_6597_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B),B2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,B2),F4)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vb(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,divide_divide(B,A2,B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_6598_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vc(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_6599_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vc(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_6600_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G3: fun(A,B),A2: B,F4: filter(A),F3: fun(A,B)] :
          ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vd(fun(A,B),fun(fun(A,B),fun(A,B)),G3),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform
tff(fact_6601_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ve(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform2
tff(fact_6602_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vc(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_6603_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),G3: fun(A,B),F4: filter(A),A2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ve(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
          <=> filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_6604_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vf(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_6605_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vg(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_6606_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vh(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_6607_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vi(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_6608_continuous__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,B,F4,G3)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_vj(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_mult
tff(fact_6609_continuous__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,B,F4,G3)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_vk(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_mult'
tff(fact_6610_continuous__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F3: fun(A,B),C3: B] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => topolo3448309680560233919inuous(A,B,F4,aa(B,fun(A,B),aTP_Lamp_vl(fun(A,B),fun(B,fun(A,B)),F3),C3)) ) ) ).

% continuous_mult_left
tff(fact_6611_continuous__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F4: filter(A),F3: fun(A,B),C3: B] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => topolo3448309680560233919inuous(A,B,F4,aa(B,fun(A,B),aTP_Lamp_vm(fun(A,B),fun(B,fun(A,B)),F3),C3)) ) ) ).

% continuous_mult_right
tff(fact_6612_tendsto__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B),B2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)),F4) ) ) ) ).

% tendsto_mult
tff(fact_6613_tendsto__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vo(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),C3),L)),F4) ) ) ).

% tendsto_mult_left
tff(fact_6614_tendsto__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vp(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),L),C3)),F4) ) ) ).

% tendsto_mult_right
tff(fact_6615_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,one_one(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vq(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,one_one(B)),F4) ) ) ) ).

% tendsto_mult_one
tff(fact_6616_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,B,F4,G3)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_vr(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_add
tff(fact_6617_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B),B2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vf(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),F4) ) ) ) ).

% tendsto_add
tff(fact_6618_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C3: B,F3: fun(A,B),D3: B,F4: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vs(B,fun(fun(A,B),fun(A,B)),C3),F3),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),C3),D3)),F4)
        <=> filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,D3),F4) ) ) ).

% tendsto_add_const_iff
tff(fact_6619_tendsto__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B),B2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,B2),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vt(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),A2),B2)),F4) ) ) ) ).

% tendsto_diff
tff(fact_6620_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,B,F4,G3)
           => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_vu(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_diff
tff(fact_6621_tendsto__arcosh,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
       => filterlim(A,real,aTP_Lamp_vv(fun(A,real),fun(A,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ).

% tendsto_arcosh
tff(fact_6622_tendsto__log,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A),G3: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( aa(real,$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_vw(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_6623_IVT,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F3: fun(B,A),A2: B,Y: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,A2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F3,B2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( ! [X4: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X4,top_top(set(B))),F3) )
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F3,X4) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_6624_IVT2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F3: fun(B,A),B2: B,Y: A,A2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,B2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F3,A2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( ! [X4: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X4,top_top(set(B))),F3) )
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F3,X4) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_6625_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [S: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S)
                & ! [X: A] :
                    ( ( ( X != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),S) )
                   => aa(real,$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,F3,X)),L5))),R2) ) ) ) ) ) ).

% LIM_D
tff(fact_6626_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F3: fun(A,B),L5: B] :
          ( ! [R4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X4),A2))),S7) )
                     => aa(real,$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,F3,X4)),L5))),R4) ) ) )
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_6627_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A2))),S6) )
                     => aa(real,$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,F3,X3)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_6628_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A2: A,F3: fun(A,B),G3: fun(A,B),L: B] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => ( ! [X4: A] :
                ( ( X4 != A2 )
               => ( aa(real,$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))),R)
                 => ( aa(A,B,F3,X4) = aa(A,B,G3,X4) ) ) )
           => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_6629_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,A),A2: A,D4: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_vx(fun(A,A),fun(A,fun(A,A)),F3),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_vy(fun(A,A),fun(A,fun(A,A)),F3),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_6630_LIM__fun__less__zero,axiom,
    ! [F3: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [R4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
            & ! [X: real] :
                ( ( ( X != C3 )
                  & 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),C3),X))),R4) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,X)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_6631_LIM__fun__not__zero,axiom,
    ! [F3: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
            & ! [X: real] :
                ( ( ( X != C3 )
                  & 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),C3),X))),R4) )
               => ( aa(real,real,F3,X) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_6632_LIM__fun__gt__zero,axiom,
    ! [F3: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [R4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
            & ! [X: real] :
                ( ( ( X != C3 )
                  & 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),C3),X))),R4) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F3,X)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_6633_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),B2: B,A2: A,G3: fun(B,C),C3: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G3,topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & aa(real,$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))),D6) )
                     => ( aa(A,B,F3,X4) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_up(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_6634_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,S2: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),G3)
           => ( ( aa(A,B,G3,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S2),aa(fun(A,B),fun(A,B),aTP_Lamp_vz(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_6635_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A2: A,F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_vj(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% isCont_mult
tff(fact_6636_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A2: A,F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_vr(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% isCont_add
tff(fact_6637_isCont__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_wa(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% isCont_diff
tff(fact_6638_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,F3: fun(A,B),Nb: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_uv(fun(A,B),fun(nat,fun(A,B)),F3),Nb)) ) ) ).

% isCont_power
tff(fact_6639_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wb(fun(A,A),fun(A,fun(A,A)),F3),Xb),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_6640_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wb(fun(A,A),fun(A,fun(A,A)),F3),Xb),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_6641_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_wc(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_6642_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K2: real,F3: fun(A,B),K5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
         => ( ! [H3: A] :
                ( ( H3 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,H3))),aa(real,real,aa(real,fun(real,real),times_times(real),K5),real_V7770717601297561774m_norm(A,H3))) ) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_6643_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F3) )
           => ? [M7: A] :
                ( ! [X: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M7),aa(real,A,F3,X)) )
                & ? [X4: real] :
                    ( aa(real,$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,A,F3,X4) = M7 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_6644_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F3) )
           => ? [M7: A] :
                ( ! [X: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F3,X)),M7) )
                & ? [X4: real] :
                    ( aa(real,$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,A,F3,X4) = M7 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_6645_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F3) )
           => ? [M7: A] :
              ! [X: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F3,X)),M7) ) ) ) ) ).

% isCont_bounded
tff(fact_6646_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,Xb: A] :
          ( has_derivative(A,A,F3,aa(A,fun(A,A),times_times(A),D4),topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wb(fun(A,A),fun(A,fun(A,A)),F3),Xb),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_6647_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => ( ( aa(A,B,G3,A2) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_vz(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% isCont_divide
tff(fact_6648_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),F4: filter(B),A2: A] :
          ( filterlim(A,B,F3,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_wd(fun(A,B),fun(A,fun(A,B)),F3),A2),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_6649_CARAT__DERIV,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A,Xb: A] :
          ( has_field_derivative(A,F3,L,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ? [G6: fun(A,A)] :
              ( ! [Z2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,F3,Z2)),aa(A,A,F3,Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G6,Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),Xb))
              & topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),G6)
              & ( aa(A,A,G6,Xb) = L ) ) ) ) ).

% CARAT_DERIV
tff(fact_6650_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( ! [X4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X4),B2) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X4,top_top(set(real))),F3) )
           => ? [M7: A] :
                ( ! [X: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F3,X)),M7) )
                & ! [N6: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N6),M7)
                   => ? [X4: real] :
                        ( aa(real,$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(A),N6),aa(real,A,F3,X4)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_6651_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),F4: filter(B),A2: A,D3: A] :
          ( filterlim(A,B,F3,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F3),aa(A,fun(A,A),plus_plus(A),D3)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_6652_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),D3: A,F4: filter(B),A2: A] :
          ( filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F3),aa(A,fun(A,A),plus_plus(A),D3)),F4,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),D3),top_top(set(A))))
        <=> filterlim(A,B,F3,F4,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_6653_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: real,A2: fun(nat,A),F3: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S2)
         => ( ! [X4: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X4)),S2)
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),A2),X4),aa(A,A,F3,X4)) )
           => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_6654_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S2: real,A2: fun(nat,A),F3: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S2)
         => ( ! [X4: A] :
                ( ( X4 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X4)),S2)
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),A2),X4),aa(A,A,F3,X4)) ) )
           => filterlim(A,A,F3,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_6655_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K2: real,F3: fun(nat,real),G3: fun(A,fun(nat,B))] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K2)
         => ( summable(real,F3)
           => ( ! [H3: A,N: nat] :
                  ( ( H3 != zero_zero(A) )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G3,H3),N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F3,N)),real_V7770717601297561774m_norm(A,H3))) ) )
             => filterlim(A,B,aTP_Lamp_we(fun(A,fun(nat,B)),fun(A,B),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_6656_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_6657_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_wf(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(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_6658_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A,C3: fun(nat,A),Nb: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_wg(fun(nat,A),fun(nat,fun(A,A)),C3),Nb)) ) ).

% isCont_polynom
tff(fact_6659_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_6660_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_6661_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),Xb: A] :
          ( ! [Y3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C3),Y3))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_rr(fun(nat,A),fun(A,A),C3)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_6662_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_6663_isCont__inverse__function,axiom,
    ! [D3: real,Xb: real,G3: fun(real,real),F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
     => ( ! [Z3: 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),Z3),Xb))),D3)
           => ( aa(real,real,G3,aa(real,real,F3,Z3)) = Z3 ) )
       => ( ! [Z3: 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),Z3),Xb))),D3)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F3) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F3,Xb),top_top(set(real))),G3) ) ) ) ).

% isCont_inverse_function
tff(fact_6664_GMVT_H,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),G3: fun(real,real),G5: fun(real,real),F6: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [Z3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),B2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),F3) ) )
       => ( ! [Z3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Z3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z3),B2)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z3,top_top(set(real))),G3) ) )
         => ( ! [Z3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
                 => has_field_derivative(real,G3,aa(real,real,G5,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
           => ( ! [Z3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
                   => has_field_derivative(real,F3,aa(real,real,F6,Z3),topolo174197925503356063within(real,Z3,top_top(set(real)))) ) )
             => ? [C2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C2),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,F3,B2)),aa(real,real,F3,A2))),aa(real,real,G5,C2)) = 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,G3,B2)),aa(real,real,G3,A2))),aa(real,real,F6,C2)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_6665_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A2: A,F3: fun(A,B),C3: fun(nat,B),K5: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_wh(fun(nat,B),fun(B,fun(nat,B)),C3),K5))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,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_wj(fun(A,B),fun(fun(nat,B),fun(A,B)),F3),C3)) ) ) ) ) ).

% isCont_powser'
tff(fact_6666_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),C3),K5))
         => ( aa(real,$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_rr(fun(nat,A),fun(A,A),C3)) ) ) ) ).

% isCont_powser
tff(fact_6667_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] : member(real,suminf(real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N8)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N8))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_6668_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] : member(real,suminf(real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N8))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N8)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_6669_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wl(A,fun(fun(nat,A),fun(nat,A)),C3),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_6670_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wm(A,fun(fun(nat,A),fun(nat,A)),C3),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_6671_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wn(A,fun(fun(nat,A),fun(nat,A)),C3),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_6672_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),L: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_wo(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_6673_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_wo(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_6674_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),K2: nat,A2: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wp(fun(nat,A),fun(nat,fun(nat,A)),F3),K2),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_6675_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),A2: A,K2: nat] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wp(fun(nat,A),fun(nat,fun(nat,A)),F3),K2),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_6676_seq__offset__neg,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),L: A,K2: nat] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wq(fun(nat,A),fun(nat,fun(nat,A)),F3),K2),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_6677_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_6678_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_6679_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),Xb: A,A2: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N6: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_6680_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),Xb: A,A2: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N6: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,X6,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xb) ) ) ) ).

% LIMSEQ_le_const
tff(fact_6681_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,N3: nat,C5: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),aa(nat,A,F3,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),L) ) ) ) ).

% Lim_bounded2
tff(fact_6682_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,M6: nat,C5: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M6),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N)),C5) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),C5) ) ) ) ).

% Lim_bounded
tff(fact_6683_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),Xb: A,Y6: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N6: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),aa(nat,A,Y6,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ) ).

% LIMSEQ_le
tff(fact_6684_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N3: nat,X6: fun(nat,A),Y6: fun(nat,A),Xb: A,Y: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),aa(nat,A,Y6,N)) )
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
           => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ) ).

% lim_mono
tff(fact_6685_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S2: set(A),A2: A] :
          ( ! [N: nat] : member(A,aa(nat,A,B2,N),S2)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2)) ) ) ) ).

% Sup_lim
tff(fact_6686_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S2: set(A),A2: A] :
          ( ! [N: nat] : member(A,aa(nat,A,B2,N),S2)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2) ) ) ) ).

% Inf_lim
tff(fact_6687_mult__nat__left__at__top,axiom,
    ! [C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_6688_mult__nat__right__at__top,axiom,
    ! [C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => filterlim(nat,nat,aTP_Lamp_wr(nat,fun(nat,nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_6689_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_ws(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_6690_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_6691_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_wt(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_6692_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X6: fun(nat,A),Xb: A,L: nat] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wu(fun(nat,A),fun(nat,fun(nat,A)),X6),L),topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_6693_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => summable(A,aTP_Lamp_wv(fun(nat,A),fun(nat,A),F3)) ) ) ).

% telescope_summable
tff(fact_6694_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => summable(A,aTP_Lamp_ww(fun(nat,A),fun(nat,A),F3)) ) ) ).

% telescope_summable'
tff(fact_6695_nested__sequence__unique,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N)),aa(nat,real,F3,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,G3,aa(nat,nat,suc,N))),aa(nat,real,G3,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N)),aa(nat,real,G3,N))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_wx(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F3),G3),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,F3,N8)),L3)
                & filterlim(nat,real,F3,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,G3,N8))
                & filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_6696_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun(nat,real)] :
      ( ! [R4: 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),R4),aa(nat,real,X6,N)) )
     => filterlim(nat,real,aTP_Lamp_wy(fun(nat,real),fun(nat,real),X6),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_6697_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_wz(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_6698_LIMSEQ__root__const,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
     => filterlim(nat,real,aTP_Lamp_xa(real,fun(nat,real),C3),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_6699_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_xb(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_6700_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_xc(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_6701_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S2: set(A),F3: fun(A,real),G3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S2),F3)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S2),G3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,A2))
             => ( ( aa(A,real,F3,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S2),aa(fun(A,real),fun(A,real),aTP_Lamp_xd(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_6702_increasing__LIMSEQ,axiom,
    ! [F3: fun(nat,real),L: real] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N)),aa(nat,real,F3,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,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,F3,N8)),E2)) )
         => filterlim(nat,real,F3,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_6703_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_xe(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_6704_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_xf(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_6705_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_xg(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_6706_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_6707_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_wv(fun(nat,A),fun(nat,A),F3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_6708_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_ww(fun(nat,A),fun(nat,A),F3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),C3)) ) ) ).

% telescope_sums'
tff(fact_6709_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_xh(real,fun(real,fun(nat,real)),Xb),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_6710_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C3)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),C3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_6711_LIMSEQ__abs__realpow__zero,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C3)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),C3)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_6712_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_xi(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_6713_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),Xb: real] :
          ( filterlim(nat,real,aTP_Lamp_xj(fun(nat,A),fun(nat,real),F3),topolo7230453075368039082e_nhds(real,Xb),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xb),one_one(real))
           => summable(A,F3) ) ) ) ).

% root_test_convergence
tff(fact_6714_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_xk(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_6715_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F3: fun(A,real),G3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,A2))
             => ( ( aa(A,real,F3,A2) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,A2))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_xd(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_6716_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$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,X6,N4)),L5))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_6717_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$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,X6,N)),L5))),R4) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_6718_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No3: nat] :
              ! [N8: nat] :
                ( aa(nat,$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,X6,N8)),L5))),R2) ) ) ) ) ).

% LIMSEQ_D
tff(fact_6719_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_6720_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim(nat,real,aTP_Lamp_xl(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,exp(real,Xb)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_6721_tendsto__at__iff__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: B,Xb: A,S2: set(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),topolo174197925503356063within(A,Xb,S2))
        <=> ! [X7: fun(nat,A)] :
              ( ! [I3: nat] : member(A,aa(nat,A,X7,I3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),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,F3),X7),topolo7230453075368039082e_nhds(B,A2),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_6722_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F3: fun(A,nat),F4: filter(A),Xb: B] :
          ( filterlim(A,nat,F3,at_top(nat),F4)
         => ( aa(real,$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_xm(fun(A,nat),fun(B,fun(A,B)),F3),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_6723_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_xn(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_6724_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N))),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_6725_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_wk(fun(nat,real),fun(nat,real),A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_6726_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Df: A,Z: A,S2: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F3,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N: nat] : aa(nat,A,S2,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_xo(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F3),Z),S2),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_6727_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_xp(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_6728_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_xq(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_6729_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_wk(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_6730_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_xr(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_xs(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_6731_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_xt(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_xs(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_6732_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_xu(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_6733_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_ck(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_6734_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_wk(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)))),suminf(real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_6735_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_xu(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_6736_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_wk(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N8)))),L3)
              & filterlim(nat,real,aTP_Lamp_xu(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_wk(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N8)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_xv(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L3),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_6737_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_xv(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_6738_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_wk(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Nb)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_6739_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_xv(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_6740_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_xw(fun(nat,A),fun(nat,A),F3),F4,at_top(nat))
    <=> filterlim(nat,A,F3,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_6741_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_xx(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F3),G3),at_top(nat))
         => ( filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_bounded_partials
tff(fact_6742_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_xy(fun(nat,$o),fun(nat,$o),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_6743_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K2: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_xz(fun(nat,$o),fun(nat,fun(nat,$o)),P),K2),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_6744_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ya(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_6745_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F3: fun(A,B),P: fun(B,$o),G3: fun(B,A)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F3,aa(B,A,G3,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G3,X4)) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F3,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_6746_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,at_top(B),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_yb(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3),F4)
           => filterlim(A,B,G3,at_top(B),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_6747_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C3),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yc(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_6748_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_yc(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ).

% filterlim_at_top
tff(fact_6749_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_6750_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,$o),ord_less_eq(A),C3),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_6751_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X4)
             => aa(A,$o,P,X4) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_6752_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_6753_eventually__sequentiallyI,axiom,
    ! [C3: nat,P: fun(nat,$o)] :
      ( ! [X4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C3),X4)
         => aa(nat,$o,P,X4) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_6754_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_6755_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_yd(fun(A,$o),fun(A,$o),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_6756_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_6757_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K2: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_xz(fun(nat,$o),fun(nat,fun(nat,$o)),P),K2),at_top(nat)) ) ).

% sequentially_offset
tff(fact_6758_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C3: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),C3),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_6759_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_qn(fun(A,fun(B,$o)),fun(A,$o),P),F4)
    <=> ? [Y7: fun(A,B)] : eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ye(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),P),Y7),F4) ) ).

% eventually_ex
tff(fact_6760_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)))
          <=> ? [B6: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B6),top_top(A))
                & ! [Z2: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B6),Z2)
                   => aa(A,$o,P,Z2) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_6761_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xb,set_ord_lessThan(A,Xb)))
        <=> ? [B6: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B6),Xb)
              & ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B6),Y5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xb)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_6762_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,set_ord_lessThan(A,Xb)))
          <=> ? [B6: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B6),Xb)
                & ! [Y5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B6),Y5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xb)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_6763_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C3),Z6)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yf(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_6764_tendsto__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),G3: fun(A,B),Net: filter(A),H: fun(A,B),C3: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_yg(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_yg(fun(A,B),fun(fun(A,B),fun(A,$o)),G3),H),Net)
           => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),Net)
             => ( filterlim(A,B,H,topolo7230453075368039082e_nhds(B,C3),Net)
               => filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,C3),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_6765_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F3: fun(A,B),Xb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_yh(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xb),A2) ) ) ) ) ).

% tendsto_upperbound
tff(fact_6766_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F3: fun(A,B),Xb: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xb),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_yi(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),Xb) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_6767_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F4: filter(A),F3: fun(A,B),Xb: B,G3: fun(A,B),Y: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xb),F4)
           => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,Y),F4)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_yj(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3),F4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Xb) ) ) ) ) ) ).

% tendsto_le
tff(fact_6768_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),Xb: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xb),F4)
        <=> ( ! [L4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L4),Xb)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yk(fun(A,B),fun(B,fun(A,$o)),F3),L4),F4) )
            & ! [U4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xb),U4)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,$o)),F3),U4),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_6769_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F3: fun(B,A),F4: filter(B)] :
          ( ! [A4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_ym(fun(B,A),fun(A,fun(B,$o)),F3),A4),F4) )
         => ( ! [A4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A4)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_yn(fun(B,A),fun(A,fun(B,$o)),F3),A4),F4) )
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,Y),F4) ) ) ) ).

% order_tendstoI
tff(fact_6770_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yk(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_6771_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),A2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_6772_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( member(A,X4,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X4) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_6773_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,$o),aTP_Lamp_yo(fun(A,$o),fun(A,fun(A,$o)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_6774_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_yp(fun(A,B),fun(B,fun(A,$o)),F3),L),F4)
         => ( ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X4),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yk(fun(A,B),fun(B,fun(A,$o)),F3),X4),F4) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_6775_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F3: fun(A,B),F4: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_yq(B,fun(fun(A,B),fun(A,$o)),L),F3),F4)
         => ( ! [X4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X4)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,$o)),F3),X4),F4) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_6776_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F3)
         => ( eventually(A,aTP_Lamp_yr(fun(A,real),fun(A,$o),F3),F4)
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_ys(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_6777_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,$o)),F3),L5),F4)
           => filterlim(A,B,F3,topolo174197925503356063within(B,L5,set_ord_lessThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_6778_tendsto__arcosh__strong,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),A2)
       => ( eventually(A,aTP_Lamp_yt(fun(A,real),fun(A,$o),F3),F4)
         => filterlim(A,real,aTP_Lamp_vv(fun(A,real),fun(A,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_6779_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F3: fun(A,B),P: fun(B,$o),G3: fun(B,A),A2: A] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F3,aa(B,A,G3,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G3,X4)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)))
               => ( ! [B4: A] :
                      ( aa(A,$o,Q,B4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),A2) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F3,at_top(B),topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_6780_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,C),K5: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_yu(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F3),G3),K5),F4)
           => filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_6781_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),A5: set(B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_yv(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F3),C3),A5),F4)
           => filterlim(A,B,F3,topolo174197925503356063within(B,C3,A5),F4) ) ) ) ).

% filterlim_at_withinI
tff(fact_6782_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yw(fun(A,B),fun(B,fun(A,$o)),F3),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_6783_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_yx(fun(A,B),fun(B,fun(A,$o)),F3),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_6784_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_6785_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_yy(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F3),G3),at_top(nat))
         => ( filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_Cauchy'
tff(fact_6786_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C3: fun(nat,A),K2: nat,Nb: nat,B5: real] :
          ( ( aa(nat,A,C3,K2) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_yz(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C3),Nb),B5),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_6787_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F3,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_za(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F6),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_6788_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,at_top(real),F4)
     => ( filterlim(A,real,G3,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_6789_real__bounded__linear,axiom,
    ! [F3: fun(real,real)] :
      ( real_V3181309239436604168linear(real,real,F3)
    <=> ? [C4: real] :
        ! [X3: real] : aa(real,real,F3,X3) = aa(real,real,aa(real,fun(real,real),times_times(real),X3),C4) ) ).

% real_bounded_linear
tff(fact_6790_bounded__linear__sub,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ( real_V3181309239436604168linear(A,B,G3)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ss(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% bounded_linear_sub
tff(fact_6791_bounded__linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ( real_V3181309239436604168linear(A,B,G3)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_st(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% bounded_linear_add
tff(fact_6792_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_zc(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_6793_bounded__linear__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G3: fun(A,B),Xb: B] :
          ( real_V3181309239436604168linear(A,B,G3)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_su(fun(A,B),fun(B,fun(A,B)),G3),Xb)) ) ) ).

% bounded_linear_const_mult
tff(fact_6794_bounded__linear__mult__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G3: fun(A,B),Y: B] :
          ( real_V3181309239436604168linear(A,B,G3)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_sv(fun(A,B),fun(B,fun(A,B)),G3),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_6795_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_6796_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_zd(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_6797_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K9: real] :
            ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K9)) ) ) ).

% bounded_linear.bounded
tff(fact_6798_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K9: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),K9)
              & ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K9)) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_6799_filterlim__pow__at__top,axiom,
    ! [A: $tType,Nb: nat,F3: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(A,real,F3,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uj(nat,fun(fun(A,real),fun(A,real)),Nb),F3),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_6800_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_6801_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B),C3: B] :
          ( filterlim(A,B,F3,at_infinity(B),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,C3),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ze(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_6802_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( filterlim(A,B,G3,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ze(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),at_infinity(B),F4) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_6803_artanh__real__at__left__1,axiom,
    filterlim(real,real,artanh(real),at_top(real),topolo174197925503356063within(real,one_one(real),set_ord_lessThan(real,one_one(real)))) ).

% artanh_real_at_left_1
tff(fact_6804_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K9: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K9)
              & ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K9)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_6805_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),K5: real] :
          ( ! [X4: A,Y3: A] : aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3))
         => ( ! [R4: real,X4: A] : aa(A,B,F3,real_V8093663219630862766scaleR(A,R4,X4)) = real_V8093663219630862766scaleR(B,R4,aa(A,B,F3,X4))
           => ( ! [X4: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K5))
             => real_V3181309239436604168linear(A,B,F3) ) ) ) ) ).

% bounded_linear_intro
tff(fact_6806_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zf(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_6807_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_6808_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( ( C3 != zero_zero(B) )
           => ( filterlim(A,B,G3,at_infinity(B),F4)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zg(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),at_infinity(B),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_6809_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( filterlim(A,B,G3,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zh(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_6810_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),Nb: nat] :
          ( filterlim(A,B,F3,at_infinity(B),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_zi(fun(A,B),fun(nat,fun(A,B)),F3),Nb),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_6811_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim(real,real,aTP_Lamp_zj(real,fun(real,real),Xb),topolo7230453075368039082e_nhds(real,exp(real,Xb)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_6812_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),C3: A,F4: filter(A),G3: fun(A,A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,C3),F4)
         => ( filterlim(A,A,G3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C3 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_qx(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_6813_filterlim__tan__at__left,axiom,
    filterlim(real,real,tan(real),at_top(real),topolo174197925503356063within(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),set_ord_lessThan(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% filterlim_tan_at_left
tff(fact_6814_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),Xb: A,S2: set(A)] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S2))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_zk(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F3),F6),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_6815_tendsto__arctan__at__top,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(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_6816_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_6817_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F6: fun(A,B),Xb: A,F3: fun(A,B),S2: 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_zl(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F6),Xb),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S2))
           => has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_derivativeI
tff(fact_6818_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),Xb: A,S2: set(A)] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S2))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_zm(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F6),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_derivative_at_within
tff(fact_6819_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_zn(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_6820_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & ? [E4: fun(A,B)] :
                ( ! [H5: A] : aa(A,B,F3,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,F3,Xb)),aa(A,B,F6,H5))),aa(A,B,E4,H5))
                & filterlim(A,real,aTP_Lamp_zo(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_6821_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),Xb: A,S2: set(A)] :
          ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S2))
        <=> ( real_V3181309239436604168linear(A,B,F6)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_za(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F6),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% has_derivative_within
tff(fact_6822_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),D4: fun(A,B),Xb: A] :
          ( has_derivative(A,B,F3,D4,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D4)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_zp(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F3),D4),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_6823_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F6: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F3,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_zr(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F3),F6),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_6824_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xb: A,S3: set(A),F3: fun(A,B),F6: fun(A,B)] :
          ( member(A,Xb,S3)
         => ( topolo1002775350975398744n_open(A,S3)
           => ( has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S3))
            <=> ( real_V3181309239436604168linear(A,B,F6)
                & ? [E4: fun(A,B)] :
                    ( ! [H5: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H5),S3)
                       => ( aa(A,B,F3,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,F3,Xb)),aa(A,B,F6,H5))),aa(A,B,E4,H5)) ) )
                    & filterlim(A,real,aTP_Lamp_zo(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_6825_open__diagonal__complement,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_zs(product_prod(A,A),$o))) ) ).

% open_diagonal_complement
tff(fact_6826_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A5: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A5)
         => ( ! [X4: A] :
                ( member(A,X4,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Xb) )
           => ~ member(A,aa(set(A),A,complete_Sup_Sup(A),A5),A5) ) ) ) ).

% Sup_notin_open
tff(fact_6827_open__Collect__ex,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,fun(B,$o))] :
          ( ! [I2: A] : topolo1002775350975398744n_open(B,aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),P,I2)))
         => topolo1002775350975398744n_open(B,aa(fun(B,$o),set(B),collect(B),aTP_Lamp_zt(fun(A,fun(B,$o)),fun(B,$o),P))) ) ) ).

% open_Collect_ex
tff(fact_6828_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A5: set(A),Xb: A] :
          ( topolo1002775350975398744n_open(A,A5)
         => ( ! [X4: A] :
                ( member(A,X4,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X4) )
           => ~ member(A,aa(set(A),A,complete_Inf_Inf(A),A5),A5) ) ) ) ).

% Inf_notin_open
tff(fact_6829_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),Xb: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( member(A,Xb,S3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B4)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Xb,B4)),S3) ) ) ) ) ) ).

% open_right
tff(fact_6830_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S3: set(A),Xb: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( member(A,Xb,S3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xb)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),Xb)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B4,Xb)),S3) ) ) ) ) ) ).

% open_left
tff(fact_6831_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_zu(product_prod(A,A),$o))) ) ).

% open_superdiagonal
tff(fact_6832_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_zv(product_prod(A,A),$o))) ) ).

% open_subdiagonal
tff(fact_6833_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S8: set(A)] :
              ( topolo1002775350975398744n_open(A,S8)
             => ( member(A,F0,S8)
               => ? [N5: nat] :
                  ! [N4: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N4)
                   => member(A,aa(nat,A,F3,N4),S8) ) ) ) ) ) ).

% lim_explicit
tff(fact_6834_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F4: filter(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F4,F3)
         => ( topolo3448309680560233919inuous(A,B,F4,G3)
           => ( ( aa(A,B,G3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_zw(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F4,aa(fun(A,B),fun(A,B),aTP_Lamp_vz(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% continuous_divide
tff(fact_6835_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A,S3: set(A),T2: set(A),U3: set(A)] :
          ( member(A,Xb,S3)
         => ( topolo1002775350975398744n_open(A,S3)
           => ( ( 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),S3)),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),S3)),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_6836_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_zw(A,A))))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_ys(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_arcosh
tff(fact_6837_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,S3: set(B),F3: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( member(B,A2,S3)
           => ( topolo1002775350975398744n_open(B,S3)
             => ( filterlim(B,C,F3,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,S3))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_uq(B,fun(fun(B,C),fun(B,C)),A2),F3),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_6838_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F3: fun(A,real),G3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F3)
         => ( topolo3448309680560233919inuous(A,real,F4,G3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_zw(A,A))))
             => ( ( aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_zw(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_zw(A,A))))
                 => topolo3448309680560233919inuous(A,real,F4,aa(fun(A,real),fun(A,real),aTP_Lamp_xd(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_6839_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F4: filter(A),F3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F4,F3)
         => ( member(real,aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F4,aTP_Lamp_zw(A,A))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
           => topolo3448309680560233919inuous(A,real,F4,aTP_Lamp_zx(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_artanh
tff(fact_6840_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E3: real,F6: fun(A,B),S2: set(A),Xb: A,F3: fun(A,B),H6: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => ( real_V3181309239436604168linear(A,B,F6)
           => ( ! [Y3: A] :
                  ( member(A,Y3,S2)
                 => ( ( Y3 != Xb )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y3,Xb)),E3)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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,F3,Y3)),aa(A,B,F3,Xb))),aa(A,B,F6,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xb)))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xb)))),aa(A,real,H6,Y3)) ) ) )
             => ( filterlim(A,real,H6,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S2))
               => has_derivative(A,B,F3,F6,topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_6841_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F3: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F3,at_bot(real),F4)
       => ( 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_zy(nat,fun(fun(real,real),fun(real,real)),Nb),F3),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_6842_dist__add__cancel2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [B2: A,A2: A,C3: 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),C3),A2)) = real_V557655796197034286t_dist(A,B2,C3) ) ).

% dist_add_cancel2
tff(fact_6843_dist__add__cancel,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C3: 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),C3)) = real_V557655796197034286t_dist(A,B2,C3) ) ).

% dist_add_cancel
tff(fact_6844_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_6845_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_6846_dist__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: real,A2: A,Y: real] : real_V557655796197034286t_dist(A,real_V8093663219630862766scaleR(A,Xb,A2),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_6847_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_6848_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_6849_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_6850_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [A2: A,B2: A,C3: A] : aa(real,$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,C3)))),real_V557655796197034286t_dist(A,A2,C3)) ) ).

% abs_dist_diff_le
tff(fact_6851_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_6852_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_6853_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_ox(A,fun(A,$o)),C3),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_6854_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C3: A] : eventually(A,aTP_Lamp_zz(A,fun(A,$o),C3),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_6855_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E4: real] :
              ( aa(real,$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,X6,M5),aa(nat,A,X6,N4))),E4) ) ) ) ) ) ).

% Cauchy_def
tff(fact_6856_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S2)
        <=> ! [E4: real] :
              ( aa(real,$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,S2,N4),aa(nat,A,S2,N5))),E4) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_6857_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => ? [M7: nat] :
              ! [M2: nat] :
                ( aa(nat,$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,X6,M2),aa(nat,A,X6,N8))),E3) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_6858_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$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,X6,M),aa(nat,A,X6,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% metric_CauchyI
tff(fact_6859_dist__of__int,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Mb: int,Nb: int] : real_V557655796197034286t_dist(A,aa(int,A,ring_1_of_int(A),Mb),aa(int,A,ring_1_of_int(A),Nb)) = aa(int,real,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_6860_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Y: A,E3: real,X2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,Y)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Y)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),E3) ) ) ) ).

% dist_triangle_half_l
tff(fact_6861_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X1: A,E3: real,X2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X1)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X2)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),E3) ) ) ) ).

% dist_triangle_half_r
tff(fact_6862_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,X2: A,E3: real,X32: A,X42: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,X32)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),divide_divide(real,E3,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X42)),E3) ) ) ) ) ).

% dist_triangle_third
tff(fact_6863_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$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,X6,M),aa(nat,A,X6,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI'
tff(fact_6864_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F3: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F3)
        <=> ! [E4: real] :
              ( aa(real,$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,F3,M5),aa(nat,A,F3,N4))),E4) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_6865_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)) = aa(int,real,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_6866_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_aaa(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ).

% filterlim_at_bot
tff(fact_6867_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z6),C3)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aaa(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_6868_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_aab(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_6869_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$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,X6,N4),L5)),R5) ) ) ) ) ).

% lim_sequentially
tff(fact_6870_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R4)
             => ? [No2: nat] :
                ! [N: nat] :
                  ( aa(nat,$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,X6,N),L5)),R4) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_6871_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No3: nat] :
              ! [N8: nat] :
                ( aa(nat,$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,X6,N8),L5)),R2) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_6872_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [J3: nat] :
            ? [M8: nat] :
            ! [M5: nat] :
              ( aa(nat,$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,X6,M5),aa(nat,A,X6,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_6873_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_6874_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G3,at_bot(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_6875_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z6: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z6),C3)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aac(fun(A,B),fun(B,fun(A,$o)),F3),Z6),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_6876_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_6877_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$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,X6,N4),L5)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_6878_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F3: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => ( filterlim(real,real,F3,at_bot(real),F4)
       => ( ~ 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_zy(nat,fun(fun(real,real),fun(real,real)),Nb),F3),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_6879_tendsto__arctan__at__bot,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,uminus_uminus(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_6880_tendsto__exp__limit__at__right,axiom,
    ! [Xb: real] : filterlim(real,real,aTP_Lamp_aad(real,fun(real,real),Xb),topolo7230453075368039082e_nhds(real,exp(real,Xb)),topolo174197925503356063within(real,zero_zero(real),set_ord_greaterThan(real,zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_6881_filterlim__tan__at__right,axiom,
    filterlim(real,real,tan(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))) ).

% filterlim_tan_at_right
tff(fact_6882_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K2: A] :
          ( member(A,I,set_ord_greaterThan(A,K2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),K2),I) ) ) ).

% greaterThan_iff
tff(fact_6883_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_greaterThan(A,Xb)),set_ord_greaterThan(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% greaterThan_subset_iff
tff(fact_6884_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),set_ord_greaterThan(A,Xb)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_6885_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_greaterThan(A,L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_6886_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xb,set_ord_greaterThan(A,Xb)))
        <=> ? [B6: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B6)
              & ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),B6)
                   => aa(A,$o,P,Y5) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_6887_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,set_ord_greaterThan(A,Xb)))
          <=> ? [B6: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B6)
                & ! [Y5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),B6)
                     => aa(A,$o,P,Y5) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_6888_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,set_ord_greaterThan(A,A2)) ) ) ) ).

% at_within_Icc_at_right
tff(fact_6889_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,set_ord_greaterThan(A,Xb))) ) ).

% eventually_at_right_less
tff(fact_6890_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,B4: A] :
              ( member(A,Xb,set_ord_lessThan(A,A4))
              & member(A,Y,set_ord_greaterThan(A,B4))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,A4)),set_ord_greaterThan(A,B4)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_6891_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,$o)] :
          ( ! [X4: A] :
              ( member(A,X4,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P,X4) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% eventually_at_rightI
tff(fact_6892_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),P2: B,F12: filter(A),C3: B,L: B] :
          ( filterlim(A,B,F3,topolo174197925503356063within(B,P2,set_ord_greaterThan(B,P2)),F12)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C3)
           => ( ( L = aa(B,B,aa(B,fun(B,B),times_times(B),C3),P2) )
             => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_aae(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo174197925503356063within(B,L,set_ord_greaterThan(B,L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_6893_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_yk(fun(A,B),fun(B,fun(A,$o)),F3),L5),F4)
           => filterlim(A,B,F3,topolo174197925503356063within(B,L5,set_ord_greaterThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_6894_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F3: fun(A,B),P: fun(B,$o),G3: fun(B,A),A2: A] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,Q,X4)
             => ( aa(A,$o,Q,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) ) ) )
         => ( ! [X4: B] :
                ( aa(B,$o,P,X4)
               => ( aa(A,B,F3,aa(B,A,G3,X4)) = X4 ) )
           => ( ! [X4: B] :
                  ( aa(B,$o,P,X4)
                 => aa(A,$o,Q,aa(B,A,G3,X4)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
               => ( ! [B4: A] :
                      ( aa(A,$o,Q,B4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B4) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F3,at_bot(B),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_6895_tendsto__arcosh__at__left__1,axiom,
    filterlim(real,real,arcosh(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,one_one(real),set_ord_greaterThan(real,one_one(real)))) ).

% tendsto_arcosh_at_left_1
tff(fact_6896_artanh__real__at__right__1,axiom,
    filterlim(real,real,artanh(real),at_bot(real),topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),one_one(real)),set_ord_greaterThan(real,aa(real,real,uminus_uminus(real),one_one(real))))) ).

% artanh_real_at_right_1
tff(fact_6897_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,G3: fun(A,B),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)),G3)
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,G3,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_aaf(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G3),F3)) ) ) ) ).

% isCont_If_ge
tff(fact_6898_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S3: set(A)] :
          ( ! [A4: A,B4: A,X4: A] :
              ( member(A,A4,S3)
             => ( member(A,B4,S3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B4)
                   => member(A,X4,S3) ) ) ) )
         => ? [A4: A,B4: A] :
              ( ( S3 = bot_bot(set(A)) )
              | ( S3 = top_top(set(A)) )
              | ( S3 = set_ord_lessThan(A,B4) )
              | ( S3 = set_ord_atMost(A,B4) )
              | ( S3 = set_ord_greaterThan(A,A4) )
              | ( S3 = set_ord_atLeast(A,A4) )
              | ( S3 = set_or5935395276787703475ssThan(A,A4,B4) )
              | ( S3 = set_or3652927894154168847AtMost(A,A4,B4) )
              | ( S3 = set_or7035219750837199246ssThan(A,A4,B4) )
              | ( S3 = set_or1337092689740270186AtMost(A,A4,B4) ) ) ) ) ).

% interval_cases
tff(fact_6899_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)
         => ( ! [F2: fun(nat,A)] :
                ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(nat,A,F2,N8))
               => ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N8)),B2)
                 => ( order_antimono(nat,A,F2)
                   => ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aag(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F2),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_6900_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K2: A] :
          ( member(A,I,set_ord_atLeast(A,K2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K2),I) ) ) ).

% atLeast_iff
tff(fact_6901_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,Xb)),set_ord_atLeast(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% atLeast_subset_iff
tff(fact_6902_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K2: A,I: A] : image(A,A,aa(A,fun(A,A),plus_plus(A),K2),set_ord_atLeast(A,I)) = set_ord_atLeast(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),K2),I)) ) ).

% image_add_atLeast
tff(fact_6903_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),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atLeast(A,L2))
        <=> ( ~ aa(A,$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_6904_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),C3),set_ord_atMost(A,B2)) = set_ord_atLeast(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ).

% image_minus_const_AtMost
tff(fact_6905_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A] : image(A,A,aa(A,fun(A,A),minus_minus(A),C3),set_ord_atLeast(A,A2)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)) ) ).

% image_minus_const_atLeast
tff(fact_6906_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K2)) = set_ord_greaterThan(nat,K2) ).

% atLeast_Suc_greaterThan
tff(fact_6907_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A5,aa(nat,nat,suc,I))),aa(nat,A,A5,I)) ) ) ).

% decseq_SucD
tff(fact_6908_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N))),aa(nat,A,X6,N))
         => order_antimono(nat,A,X6) ) ) ).

% decseq_SucI
tff(fact_6909_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_antimono(nat,A,F3)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N4))),aa(nat,A,F3,N4)) ) ) ).

% decseq_Suc_iff
tff(fact_6910_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_antimono(A,B,F3)
        <=> ! [X3: A,Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Y5)),aa(A,B,F3,X3)) ) ) ) ).

% antimono_def
tff(fact_6911_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Y3)),aa(A,B,F3,X4)) )
         => order_antimono(A,B,F3) ) ) ).

% antimonoI
tff(fact_6912_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( aa(A,$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,F3,Y)),aa(A,B,F3,Xb)) ) ) ) ).

% antimonoE
tff(fact_6913_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( aa(A,$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,F3,Y)),aa(A,B,F3,Xb)) ) ) ) ).

% antimonoD
tff(fact_6914_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_atLeast(A,L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_6915_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( order_antimono(nat,A,X6)
        <=> ! [M5: nat,N4: nat] :
              ( aa(nat,$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,X6,N4)),aa(nat,A,X6,M5)) ) ) ) ).

% decseq_def
tff(fact_6916_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),I: nat,J: nat] :
          ( order_antimono(nat,A,F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,J)),aa(nat,A,F3,I)) ) ) ) ).

% decseqD
tff(fact_6917_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,A2)),set_ord_greaterThan(A,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_6918_greaterThan__0,axiom,
    set_ord_greaterThan(nat,zero_zero(nat)) = image(nat,nat,suc,top_top(set(nat))) ).

% greaterThan_0
tff(fact_6919_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L5: A,Nb: nat] :
          ( order_antimono(nat,A,X6)
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L5),aa(nat,A,X6,Nb)) ) ) ) ).

% decseq_ge
tff(fact_6920_greaterThan__Suc,axiom,
    ! [K2: nat] : set_ord_greaterThan(nat,aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_greaterThan(nat,K2)),aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K2)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_6921_atLeast__Suc,axiom,
    ! [K2: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K2)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atLeast(nat,K2)),aa(set(nat),set(nat),insert(nat,K2),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_6922_tendsto__at__right__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,X6: fun(A,B),L5: B] :
          ( aa(A,$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_aah(fun(A,B),fun(fun(nat,A),fun(nat,B)),X6),S5),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X6,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_6923_GMVT,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),G3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [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) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X4,top_top(set(real))),F3) )
       => ( ! [X4: real] :
              ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X4)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2) )
             => differentiable(real,real,F3,topolo174197925503356063within(real,X4,top_top(set(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) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X4,top_top(set(real))),G3) )
           => ( ! [X4: real] :
                  ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X4)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2) )
                 => differentiable(real,real,G3,topolo174197925503356063within(real,X4,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( has_field_derivative(real,G3,G_c,topolo174197925503356063within(real,C2,top_top(set(real))))
                  & has_field_derivative(real,F3,F_c,topolo174197925503356063within(real,C2,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(real),C2),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,F3,B2)),aa(real,real,F3,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,G3,B2)),aa(real,real,G3,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_6924_at__within__order,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xb: A,S2: set(A)] :
          ( ( top_top(set(A)) != aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) )
         => ( topolo174197925503356063within(A,Xb,S2) = 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)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aai(A,fun(set(A),fun(A,filter(A))),Xb),S2),set_ord_greaterThan(A,Xb)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aaj(A,fun(set(A),fun(A,filter(A))),Xb),S2),set_ord_lessThan(A,Xb)))) ) ) ) ).

% at_within_order
tff(fact_6925_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C3: B,Q3: fun(A,B),Ta: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aak(B,fun(fun(A,B),fun(A,B)),C3),Q3),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C3 = zero_zero(B) )
            | differentiable(A,B,Q3,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_6926_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q3: fun(A,B),C3: B,Ta: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_aal(fun(A,B),fun(B,fun(A,B)),Q3),C3),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C3 = zero_zero(B) )
            | differentiable(A,B,Q3,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_6927_differentiable__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),Xb: A,S2: set(A),Nb: nat] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,Xb,S2))
         => differentiable(A,B,aa(nat,fun(A,B),aTP_Lamp_tl(fun(A,B),fun(nat,fun(A,B)),F3),Nb),topolo174197925503356063within(A,Xb,S2)) ) ) ).

% differentiable_power
tff(fact_6928_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( differentiable(A,B,F3,F4)
         => ( differentiable(A,B,G3,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_st(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),F4) ) ) ) ).

% differentiable_add
tff(fact_6929_differentiable__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( differentiable(A,B,F3,F4)
         => ( differentiable(A,B,G3,F4)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ss(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),F4) ) ) ) ).

% differentiable_diff
tff(fact_6930_differentiable__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: fun(A,B),Xb: A,S2: set(A),G3: fun(A,B)] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,Xb,S2))
         => ( differentiable(A,B,G3,topolo174197925503356063within(A,Xb,S2))
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sw(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% differentiable_mult
tff(fact_6931_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),Xb: A,S2: set(A),G3: fun(A,B)] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,Xb,S2))
         => ( differentiable(A,B,G3,topolo174197925503356063within(A,Xb,S2))
           => ( ( aa(A,B,G3,Xb) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tc(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo174197925503356063within(A,Xb,S2)) ) ) ) ) ).

% differentiable_divide
tff(fact_6932_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,set_ord_lessThan(A,Xb)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(A,filter(A),aTP_Lamp_aam(A,fun(A,filter(A)),Xb),set_ord_lessThan(A,Xb))) ) ) ) ).

% at_left_eq
tff(fact_6933_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,set_ord_greaterThan(A,Xb)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(A,filter(A),aTP_Lamp_aan(A,fun(A,filter(A)),Xb),set_ord_greaterThan(A,Xb))) ) ) ) ).

% at_right_eq
tff(fact_6934_at__within__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S2: set(A)] : topolo174197925503356063within(A,A2,S2) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A2)),principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ).

% at_within_def
tff(fact_6935_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xb: A,S2: set(A)] : topolo174197925503356063within(A,Xb,S2) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),image(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_aao(A,fun(set(A),fun(set(A),filter(A))),Xb),S2),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aap(A,fun(set(A),$o),Xb)))) ) ).

% at_within_eq
tff(fact_6936_MVT,axiom,
    ! [A2: real,B2: real,F3: 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),F3)
       => ( ! [X4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
               => differentiable(real,real,F3,topolo174197925503356063within(real,X4,top_top(set(real)))) ) )
         => ? [L3: real,Z3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z3)
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
              & has_field_derivative(real,F3,L3,topolo174197925503356063within(real,Z3,top_top(set(real))))
              & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,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_6937_eventually__filtercomap__at__topological,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,$o),F3: fun(A,B),A5: B,B5: set(B)] :
          ( eventually(A,P,filtercomap(A,B,F3,topolo174197925503356063within(B,A5,B5)))
        <=> ? [S8: set(B)] :
              ( topolo1002775350975398744n_open(B,S8)
              & member(B,A5,S8)
              & ! [X3: A] :
                  ( member(B,aa(A,B,F3,X3),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)),S8),B5)),aa(set(B),set(B),insert(B,A5),bot_bot(set(B)))))
                 => aa(A,$o,P,X3) ) ) ) ) ).

% eventually_filtercomap_at_topological
tff(fact_6938_continuous__on__arcosh,axiom,
    ! [A5: set(real)] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),A5),set_ord_atLeast(real,one_one(real)))
     => topolo81223032696312382ous_on(real,real,A5,arcosh(real)) ) ).

% continuous_on_arcosh
tff(fact_6939_continuous__onI__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & dense_order(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(B,A),A5: set(B)] :
          ( topolo1002775350975398744n_open(A,image(B,A,F3,A5))
         => ( ! [X4: B,Y3: B] :
                ( member(B,X4,A5)
               => ( member(B,Y3,A5)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),Y3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X4)),aa(B,A,F3,Y3)) ) ) )
           => topolo81223032696312382ous_on(B,A,A5,F3) ) ) ) ).

% continuous_onI_mono
tff(fact_6940_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S2: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( topolo81223032696312382ous_on(A,B,S2,G3)
           => ( ! [X4: A] :
                  ( member(A,X4,S2)
                 => ( aa(A,B,G3,X4) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S2,aa(fun(A,B),fun(A,B),aTP_Lamp_aaq(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ) ).

% continuous_on_divide
tff(fact_6941_continuous__on__mult__const,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [S2: set(A),C3: A] : topolo81223032696312382ous_on(A,A,S2,aa(A,fun(A,A),times_times(A),C3)) ) ).

% continuous_on_mult_const
tff(fact_6942_continuous__on__op__minus,axiom,
    ! [A: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [S2: set(A),Xb: A] : topolo81223032696312382ous_on(A,A,S2,aa(A,fun(A,A),minus_minus(A),Xb)) ) ).

% continuous_on_op_minus
tff(fact_6943_continuous__on__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [S2: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( topolo81223032696312382ous_on(A,B,S2,G3)
           => topolo81223032696312382ous_on(A,B,S2,aa(fun(A,B),fun(A,B),aTP_Lamp_aar(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_on_diff
tff(fact_6944_continuous__on__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [S2: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( topolo81223032696312382ous_on(A,B,S2,G3)
           => topolo81223032696312382ous_on(A,B,S2,aa(fun(A,B),fun(A,B),aTP_Lamp_aas(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_on_add
tff(fact_6945_continuous__on__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S2: set(A),F3: fun(A,B),C3: B] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => topolo81223032696312382ous_on(A,B,S2,aa(B,fun(A,B),aTP_Lamp_aat(fun(A,B),fun(B,fun(A,B)),F3),C3)) ) ) ).

% continuous_on_mult_right
tff(fact_6946_continuous__on__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S2: set(A),F3: fun(A,B),C3: B] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => topolo81223032696312382ous_on(A,B,S2,aa(B,fun(A,B),aTP_Lamp_aau(fun(A,B),fun(B,fun(A,B)),F3),C3)) ) ) ).

% continuous_on_mult_left
tff(fact_6947_continuous__on__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A5,F3)
         => ( topolo81223032696312382ous_on(A,B,A5,G3)
           => topolo81223032696312382ous_on(A,B,A5,aa(fun(A,B),fun(A,B),aTP_Lamp_aav(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_on_mult'
tff(fact_6948_continuous__on__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [S2: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( topolo81223032696312382ous_on(A,B,S2,G3)
           => topolo81223032696312382ous_on(A,B,S2,aa(fun(A,B),fun(A,B),aTP_Lamp_aaw(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_on_mult
tff(fact_6949_continuous__on__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S2: set(A),F3: fun(A,B),Nb: nat] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => topolo81223032696312382ous_on(A,B,S2,aa(nat,fun(A,B),aTP_Lamp_aax(fun(A,B),fun(nat,fun(A,B)),F3),Nb)) ) ) ).

% continuous_on_power
tff(fact_6950_continuous__on__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,nat)] :
          ( topolo81223032696312382ous_on(A,B,A5,F3)
         => ( topolo81223032696312382ous_on(A,nat,A5,G3)
           => topolo81223032696312382ous_on(A,B,A5,aa(fun(A,nat),fun(A,B),aTP_Lamp_aay(fun(A,B),fun(fun(A,nat),fun(A,B)),F3),G3)) ) ) ) ).

% continuous_on_power'
tff(fact_6951_continuous__on__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(A),F3: fun(A,B),G3: fun(A,C)] :
          ( topolo81223032696312382ous_on(A,B,S2,F3)
         => ( topolo81223032696312382ous_on(A,C,S2,G3)
           => topolo81223032696312382ous_on(A,product_prod(B,C),S2,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_aaz(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3)) ) ) ) ).

% continuous_on_Pair
tff(fact_6952_IVT_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F3: fun(B,A),A2: B,Y: A,B2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,A2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F3,B2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A2,B2),F3)
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F3,X4) = Y ) ) ) ) ) ) ) ).

% IVT'
tff(fact_6953_IVT2_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F3: fun(B,A),B2: B,Y: A,A2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,B2)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(B,A,F3,A2))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),B2)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A2,B2),F3)
               => ? [X4: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),X4)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X4),B2)
                    & ( aa(B,A,F3,X4) = Y ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_6954_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F3)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G3)
           => topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_aba(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3))) ) ) ) ).

% open_Collect_less
tff(fact_6955_continuous__on__arcosh_H,axiom,
    ! [A5: set(real),F3: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A5,F3)
     => ( ! [X4: real] :
            ( member(real,X4,A5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,F3,X4)) )
       => topolo81223032696312382ous_on(real,real,A5,aTP_Lamp_abb(fun(real,real),fun(real,real),F3)) ) ) ).

% continuous_on_arcosh'
tff(fact_6956_eventually__filtercomap__at__top__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F3,at_top(B)))
        <=> ? [N5: B] :
            ! [X3: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N5),aa(A,B,F3,X3))
             => aa(A,$o,P,X3) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_6957_eventually__filtercomap__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [P: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F3,at_top(B)))
        <=> ? [N5: B] :
            ! [X3: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),N5),aa(A,B,F3,X3))
             => aa(A,$o,P,X3) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_6958_eventually__filtercomap__at__bot__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F3,at_bot(B)))
        <=> ? [N5: B] :
            ! [X3: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),N5)
             => aa(A,$o,P,X3) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_6959_eventually__filtercomap__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [P: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P,filtercomap(A,B,F3,at_bot(B)))
        <=> ? [N5: B] :
            ! [X3: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X3)),N5)
             => aa(A,$o,P,X3) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_6960_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F3: fun(A,real),G3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S2,F3)
         => ( topolo81223032696312382ous_on(A,real,S2,G3)
           => ( ! [X4: A] :
                  ( member(A,X4,S2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,X4)) )
             => ( ! [X4: A] :
                    ( member(A,X4,S2)
                   => ( aa(A,real,F3,X4) != one_one(real) ) )
               => ( ! [X4: A] :
                      ( member(A,X4,S2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,X4)) )
                 => topolo81223032696312382ous_on(A,real,S2,aa(fun(A,real),fun(A,real),aTP_Lamp_abc(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_6961_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_6962_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_6963_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S2,F3)
         => ( ! [X4: A] :
                ( member(A,X4,S2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F3,X4))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F3,X4)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S2,aTP_Lamp_abd(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_on_arccos
tff(fact_6964_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S2,F3)
         => ( ! [X4: A] :
                ( member(A,X4,S2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F3,X4))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F3,X4)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S2,aTP_Lamp_abe(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_on_arcsin
tff(fact_6965_continuous__on__artanh,axiom,
    ! [A5: set(real)] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),A5),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
     => topolo81223032696312382ous_on(real,real,A5,artanh(real)) ) ).

% continuous_on_artanh
tff(fact_6966_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,B2: A,F3: fun(A,A)] :
          ( ! [X4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X4)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),B2)
               => ? [Y4: A] : has_field_derivative(A,F3,Y4,topolo174197925503356063within(A,X4,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A2,B2),F3) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_6967_continuous__on__artanh_H,axiom,
    ! [A5: set(real),F3: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A5,F3)
     => ( ! [X4: real] :
            ( member(real,X4,A5)
           => member(real,aa(real,real,F3,X4),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))) )
       => topolo81223032696312382ous_on(real,real,A5,aTP_Lamp_abf(fun(real,real),fun(real,real),F3)) ) ) ).

% continuous_on_artanh'
tff(fact_6968_mvt,axiom,
    ! [A2: real,B2: real,F3: 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),F3)
       => ( ! [X4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),X4)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X4),B2)
               => has_derivative(real,real,F3,aa(real,fun(real,real),F6,X4),topolo174197925503356063within(real,X4,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,F3,B2)),aa(real,real,F3,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_6969_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_abg(A,B)) ) ).

% continuous_on_of_int_floor
tff(fact_6970_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_abh(A,B)) ) ).

% continuous_on_of_int_ceiling
tff(fact_6971_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_6972_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_6973_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A,B2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,set_ord_greaterThan(A,A2)))
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,B2)),topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)))
           => ( ! [X4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),B2)
                   => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,X4)),topolo174197925503356063within(A,X4,top_top(set(A)))) ) )
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_6974_card__Pow,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow2(A,A5)) = 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),A5)) ) ) ).

% card_Pow
tff(fact_6975_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),$o))),aTP_Lamp_abj(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_6976_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R2: set(product_prod(A,A)),S2: set(product_prod(B,B))] :
      ( 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),A3),B3)),lex_prod(A,B,R2,S2))
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A3),R2)
        | ( ( A2 = A3 )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B3),S2) ) ) ) ).

% in_lex_prod
tff(fact_6977_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_6978_subseqs__powset,axiom,
    ! [A: $tType,Xs: list(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_6979_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),R: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R) = 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_abl(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),P),R)))) ).

% same_fst_def
tff(fact_6980_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),cons(B,Y),Ys)) = aa(list(A),list(product_prod(A,B)),case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_abm(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).

% zip_Cons
tff(fact_6981_same__fstI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Xb: A,Y8: B,Y: B,R: fun(A,set(product_prod(B,B)))] :
      ( aa(A,$o,P,Xb)
     => ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y8),Y),aa(A,set(product_prod(B,B)),R,Xb))
       => 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),Y8)),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,R)) ) ) ).

% same_fstI
tff(fact_6982_list_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,fun(list(B),A))] : aa(list(B),A,case_list(A,B,F1,F22),nil(B)) = F1 ).

% list.simps(4)
tff(fact_6983_list_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,fun(list(C),B)),List: list(C)] : aa(B,A,H,aa(list(C),B,case_list(B,C,F1,F22),List)) = aa(list(C),A,case_list(A,C,aa(B,A,H,F1),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_abn(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),H),F22)),List) ).

% list.case_distrib
tff(fact_6984_list_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,fun(list(B),A)),X21: B,X222: list(B)] : aa(list(B),A,case_list(A,B,F1,F22),aa(list(B),list(B),cons(B,X21),X222)) = aa(list(B),A,aa(B,fun(list(B),A),F22,X21),X222) ).

% list.simps(5)
tff(fact_6985_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(A),A,case_list(A,A,Xb,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_abo(A,fun(list(A),fun(A,fun(list(A),A))),Xb),Xs)),Xs) ) ).

% min_list.simps
tff(fact_6986_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),cons(A,Xb),Xs),Ys) = aa(list(B),list(product_prod(A,B)),case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_abp(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Xb),Xs)),Ys) ).

% zip_Cons1
tff(fact_6987_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K2: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
       => 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))),K2),R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abr(set(product_prod(A,A)),fun(nat,$o),R))))) ) ) ).

% relpow_finite_bounded1
tff(fact_6988_the__elem__set,axiom,
    ! [A: $tType,Xb: A] : the_elem(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,Xb),nil(A)))) = Xb ).

% the_elem_set
tff(fact_6989_relpow__1,axiom,
    ! [A: $tType,R: 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)),R) = R ).

% relpow_1
tff(fact_6990_finite__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Nb: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
       => aa(set(product_prod(A,A)),$o,finite_finite(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),R)) ) ) ).

% finite_relpow
tff(fact_6991_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> aa(list(A),$o,case_list($o,A,$false,aTP_Lamp_abs(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_6992_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> aa(list(A),$o,case_list($o,A,$true,aTP_Lamp_abt(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_6993_relpow__Suc__D2_H,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A)),X: A,Y4: A,Z4: A] :
      ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),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),R))
        & 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) )
     => ? [W2: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),W2),R)
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W2),Z4),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),R)) ) ) ).

% relpow_Suc_D2'
tff(fact_6994_relpow__0__I,axiom,
    ! [A: $tType,Xb: A,R: set(product_prod(A,A))] : 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)),R)) ).

% relpow_0_I
tff(fact_6995_relpow__0__E,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( 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)),R))
     => ( Xb = Y ) ) ).

% relpow_0_E
tff(fact_6996_relpow__Suc__I2,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A)),Z: A,Nb: nat] :
      ( 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)
     => ( 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),R))
       => 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)),R)) ) ) ).

% relpow_Suc_I2
tff(fact_6997_relpow__Suc__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( 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)),R))
     => ~ ! [Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y3),R)
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),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),R)) ) ) ).

% relpow_Suc_E2
tff(fact_6998_relpow__Suc__D2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( 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)),R))
     => ? [Y3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y3),R)
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),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),R)) ) ) ).

% relpow_Suc_D2
tff(fact_6999_relpow__Suc__I,axiom,
    ! [A: $tType,Xb: A,Y: A,Nb: nat,R: set(product_prod(A,A)),Z: A] :
      ( 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),R))
     => ( 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),R)
       => 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)),R)) ) ) ).

% relpow_Suc_I
tff(fact_7000_relpow__Suc__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( 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)),R))
     => ~ ! [Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y3),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),R))
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z),R) ) ) ).

% relpow_Suc_E
tff(fact_7001_relpowp__relpow__eq,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A)),X: A,Xa: A] :
      ( aa(A,$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_abu(set(product_prod(A,A)),fun(A,fun(A,$o)),R)),X),Xa)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),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),R)) ) ).

% relpowp_relpow_eq
tff(fact_7002_relpow__E2,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( 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),R))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y3),R)
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),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),R)) ) ) ) ) ).

% relpow_E2
tff(fact_7003_relpow__E,axiom,
    ! [A: $tType,Xb: A,Z: A,Nb: nat,R: set(product_prod(A,A))] :
      ( 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),R))
     => ( ( ( Nb = zero_zero(nat) )
         => ( Xb != Z ) )
       => ~ ! [Y3: A,M: nat] :
              ( ( Nb = aa(nat,nat,suc,M) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y3),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),R))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z),R) ) ) ) ) ).

% relpow_E
tff(fact_7004_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_7005_relpow__fun__conv,axiom,
    ! [A: $tType,A2: A,B2: A,Nb: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Nb),R))
    <=> ? [F5: fun(nat,A)] :
          ( ( aa(nat,A,F5,zero_zero(nat)) = A2 )
          & ( aa(nat,A,F5,Nb) = B2 )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Nb)
             => 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,I3)),aa(nat,A,F5,aa(nat,nat,suc,I3))),R) ) ) ) ).

% relpow_fun_conv
tff(fact_7006_relpow__finite__bounded,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K2: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,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))),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))),K2),R)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abv(set(product_prod(A,A)),fun(nat,$o),R))))) ) ).

% relpow_finite_bounded
tff(fact_7007_ntrancl__def,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,Nb,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abw(nat,fun(nat,$o),Nb)))) ).

% ntrancl_def
tff(fact_7008_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),R)
     => ( transitive_trancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abr(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_7009_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$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),R2))
     => ( ! [A4: A,B4: B] :
            ( 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),B4)),R2)
           => aa(B,$o,aa(A,fun(B,$o),P,A4),B4) )
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( 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),B4)),transitive_trancl(product_prod(A,B),R2))
             => ( 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),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R2)
               => ( aa(B,$o,aa(A,fun(B,$o),P,A4),B4)
                 => 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_7010_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( 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,R2))
     => ( ~ 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),R2)
       => ~ ! [B4: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4),transitive_trancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22),R2) ) ) ) ).

% trancl.cases
tff(fact_7011_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( 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,R2))
    <=> ( ? [A6: A,B6: A] :
            ( ( A1 = A6 )
            & ( A22 = B6 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B6),R2) )
        | ? [A6: A,B6: A,C4: A] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B6),transitive_trancl(A,R2))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B6),C4),R2) ) ) ) ).

% trancl.simps
tff(fact_7012_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( 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)
     => 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)) ) ).

% trancl.r_into_trancl
tff(fact_7013_tranclE,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( 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))
     => ( ~ 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)
       => ~ ! [C2: A] :
              ( 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))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C2),B2),R2) ) ) ) ).

% tranclE
tff(fact_7014_trancl__trans,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),Z: A] :
      ( 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))
     => ( 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,R2))
       => 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,R2)) ) ) ).

% trancl_trans
tff(fact_7015_trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$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))
     => ( ! [Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y3),R2)
           => aa(A,$o,P,Y3) )
       => ( ! [Y3: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y3),transitive_trancl(A,R2))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R2)
               => ( aa(A,$o,P,Y3)
                 => aa(A,$o,P,Z3) ) ) )
         => aa(A,$o,P,B2) ) ) ) ).

% trancl_induct
tff(fact_7016_r__r__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A)),C3: A] :
      ( 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)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R)
       => 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)) ) ) ).

% r_r_into_trancl
tff(fact_7017_converse__tranclE,axiom,
    ! [A: $tType,Xb: A,Z: A,R2: set(product_prod(A,A))] :
      ( 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,R2))
     => ( ~ 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),R2)
       => ~ ! [Y3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y3),R2)
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z),transitive_trancl(A,R2)) ) ) ) ).

% converse_tranclE
tff(fact_7018_irrefl__trancl__rD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: A,Y: A] :
      ( ! [X4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),transitive_trancl(A,R2))
     => ( 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)
       => ( Xb != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_7019_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( 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))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R2)
       => 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,R2)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_7020_trancl__into__trancl2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( 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)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),transitive_trancl(A,R2))
       => 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,R2)) ) ) ).

% trancl_into_trancl2
tff(fact_7021_trancl__trans__induct,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),P: fun(A,fun(A,$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))
     => ( ! [X4: A,Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R2)
           => aa(A,$o,aa(A,fun(A,$o),P,X4),Y3) )
       => ( ! [X4: A,Y3: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),transitive_trancl(A,R2))
             => ( aa(A,$o,aa(A,fun(A,$o),P,X4),Y3)
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),transitive_trancl(A,R2))
                 => ( aa(A,$o,aa(A,fun(A,$o),P,Y3),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),P,X4),Z3) ) ) ) )
         => aa(A,$o,aa(A,fun(A,$o),P,Xb),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_7022_converse__trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$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))
     => ( ! [Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),B2),R2)
           => aa(A,$o,P,Y3) )
       => ( ! [Y3: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R2)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2),transitive_trancl(A,R2))
               => ( aa(A,$o,P,Z3)
                 => aa(A,$o,P,Y3) ) ) )
         => aa(A,$o,P,A2) ) ) ) ).

% converse_trancl_induct
tff(fact_7023_finite__trancl__ntranl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),R)
     => ( transitive_trancl(A,R) = 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)),R)),one_one(nat)),R) ) ) ).

% finite_trancl_ntranl
tff(fact_7024_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_7025_trancl__power,axiom,
    ! [A: $tType,P2: product_prod(A,A),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),P2,transitive_trancl(A,R))
    <=> ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N4)
          & 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),R)) ) ) ).

% trancl_power
tff(fact_7026_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)
         => ( ! [F2: fun(nat,A)] :
                ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(nat,A,F2,N8))
               => ( ! [N8: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N8)),A2)
                 => ( order_mono(nat,A,F2)
                   => ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aag(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F2),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_7027_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [X3: A] :
                ! [N4: nat] : aa(real,$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,X6,N4)),aa(A,A,uminus_uminus(A),X3)))),K3) ) ) ) ).

% Bseq_iff2
tff(fact_7028_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F3: fun(A,A),P2: A,K2: nat] :
          ( order_mono(A,A,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,F3,P2))
           => aa(A,$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)),K2),F3),top_top(A))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_7029_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F3: fun(A,A),P2: A,K2: nat] :
          ( order_mono(A,A,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F3,P2)),P2)
           => aa(A,$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)),K2),F3),bot_bot(A))),P2) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_7030_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_7031_mono__Suc,axiom,
    order_mono(nat,nat,suc) ).

% mono_Suc
tff(fact_7032_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),Y) ) ) ) ).

% mono_strict_invE
tff(fact_7033_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_7034_Bseq__add__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_abx(fun(nat,A),fun(A,fun(nat,A)),F3),C3),at_top(nat))
        <=> bfun(nat,A,F3,at_top(nat)) ) ) ).

% Bseq_add_iff
tff(fact_7035_Bseq__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( bfun(nat,A,F3,at_top(nat))
         => bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_abx(fun(nat,A),fun(A,fun(nat,A)),F3),C3),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_7036_Bseq__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( bfun(nat,A,F3,at_top(nat))
         => ( bfun(nat,A,G3,at_top(nat))
           => bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aby(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G3),at_top(nat)) ) ) ) ).

% Bseq_mult
tff(fact_7037_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_dp(fun(nat,A),fun(nat,A),F3),at_top(nat))
        <=> bfun(nat,A,F3,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_7038_Bseq__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),K2: nat] :
          ( bfun(nat,A,X6,at_top(nat))
         => bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_abz(fun(nat,A),fun(nat,fun(nat,A)),X6),K2),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_7039_Bseq__offset,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),K2: nat] :
          ( bfun(nat,A,aa(nat,fun(nat,A),aTP_Lamp_abz(fun(nat,A),fun(nat,fun(nat,A)),X6),K2),at_top(nat))
         => bfun(nat,A,X6,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_7040_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F3: fun(A,B),A5: A,B5: A] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B5))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F3,A5)),aa(A,B,F3,B5))) ) ) ).

% mono_inf
tff(fact_7041_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_7042_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_mono(A,B,F3)
        <=> ! [X3: A,Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y5)) ) ) ) ).

% mono_def
tff(fact_7043_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y3)) )
         => order_mono(A,B,F3) ) ) ).

% monoI
tff(fact_7044_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F3)
         => ( aa(A,$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,F3,Xb)),aa(A,B,F3,Y)) ) ) ) ).

% monoE
tff(fact_7045_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F3)
         => ( aa(A,$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,F3,Xb)),aa(A,B,F3,Y)) ) ) ) ).

% monoD
tff(fact_7046_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),I: nat,J: nat] :
          ( order_mono(nat,A,F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,I)),aa(nat,A,F3,J)) ) ) ) ).

% incseqD
tff(fact_7047_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( order_mono(nat,A,X6)
        <=> ! [M5: nat,N4: nat] :
              ( aa(nat,$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,X6,M5)),aa(nat,A,X6,N4)) ) ) ) ).

% incseq_def
tff(fact_7048_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_mono(nat,A,F3)
        <=> ! [N4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N4)),aa(nat,A,F3,aa(nat,nat,suc,N4))) ) ) ).

% incseq_Suc_iff
tff(fact_7049_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,N)),aa(nat,A,X6,aa(nat,nat,suc,N)))
         => order_mono(nat,A,X6) ) ) ).

% incseq_SucI
tff(fact_7050_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: fun(nat,A),I: nat] :
          ( order_mono(nat,A,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A5,I)),aa(nat,A,A5,aa(nat,nat,suc,I))) ) ) ).

% incseq_SucD
tff(fact_7051_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),Xb: A,Y: A] :
          ( order_mono(A,B,F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mono_invE
tff(fact_7052_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),A5: A,B5: A,Nb: nat] :
          ( order_mono(A,A,F3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B5)
           => aa(A,$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),F3),A5)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),B5)) ) ) ) ).

% funpow_mono
tff(fact_7053_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),Nb: nat] :
          ( order_mono(A,A,F3)
         => 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),F3)) ) ) ).

% mono_pow
tff(fact_7054_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_aca(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_7055_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F3: fun(A,B),Mb: A,Nb: A,M4: B,N2: B] :
          ( order_mono(A,B,F3)
         => ( ( image(A,B,F3,set_or7035219750837199246ssThan(A,Mb,Nb)) = set_or7035219750837199246ssThan(B,M4,N2) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Mb),Nb)
             => ( aa(A,B,F3,Mb) = M4 ) ) ) ) ) ).

% mono_image_least
tff(fact_7056_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),I: nat,J: nat,Xb: A,Y: A] :
          ( order_mono(A,A,F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,F3,Xb))
               => aa(A,$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)),I),F3),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),F3),Y)) ) ) ) ) ) ).

% funpow_mono2
tff(fact_7057_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F3,A5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ).

% mono_Sup
tff(fact_7058_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A5: fun(C,A),I6: set(C)] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acb(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5),I6))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),image(C,A,A5,I6)))) ) ) ).

% mono_SUP
tff(fact_7059_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A5))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F3,A5))) ) ) ).

% mono_Inf
tff(fact_7060_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F3: fun(A,B),A5: fun(C,A),I6: set(C)] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),image(C,A,A5,I6)))),aa(set(B),B,complete_Inf_Inf(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acb(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A5),I6))) ) ) ).

% mono_INF
tff(fact_7061_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_acc(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_7062_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dj(A,fun(fun(nat,A),fun(nat,A)),C3),F3),at_top(nat))
          <=> bfun(nat,A,F3,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_7063_incseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L5: A,Nb: nat] :
          ( order_mono(nat,A,X6)
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X6,Nb)),L5) ) ) ) ).

% incseq_le
tff(fact_7064_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Mb: nat,Nb: nat,F3: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( order_mono(A,A,F3)
           => aa(A,$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),F3),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),F3),top_top(A))) ) ) ) ).

% funpow_increasing
tff(fact_7065_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Mb: nat,Nb: nat,F3: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
         => ( order_mono(A,A,F3)
           => aa(A,$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),F3),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),F3),bot_bot(A))) ) ) ) ).

% funpow_decreasing
tff(fact_7066_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N5: nat] :
            ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N5))) ) ) ).

% Bseq_iff1a
tff(fact_7067_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N5: nat] :
            ! [N4: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N4))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N5))) ) ) ).

% Bseq_iff
tff(fact_7068_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_7069_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K2)
     => order_mono(nat,nat,aTP_Lamp_acd(nat,fun(nat,nat),K2)) ) ).

% mono_ge2_power_minus_self
tff(fact_7070_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( aa(set(A),$o,finite_finite(A),image(nat,A,F3,top_top(set(nat))))
         => ( order_mono(nat,A,F3)
           => ( ! [N: nat] :
                  ( ( aa(nat,A,F3,N) = aa(nat,A,F3,aa(nat,nat,suc,N)) )
                 => ( aa(nat,A,F3,aa(nat,nat,suc,N)) = aa(nat,A,F3,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,F3,M2)),aa(nat,A,F3,N8)) ) ) )
                  & ! [N8: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N8)
                     => ( aa(nat,A,F3,N7) = aa(nat,A,F3,N8) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_7071_tendsto__at__left__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [B2: A,A2: A,X6: fun(A,B),L5: B] :
          ( aa(A,$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_aah(fun(A,B),fun(fun(nat,A),fun(nat,B)),X6),S5),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) ) ) ) )
           => filterlim(A,B,X6,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ).

% tendsto_at_left_sequentially
tff(fact_7072_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$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,X6,N4)),aa(A,A,uminus_uminus(A),aa(nat,A,X6,N5))))),K3) ) ) ) ).

% Bseq_iff3
tff(fact_7073_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F5: fun(nat,nat)] :
          ( order_mono(nat,nat,F5)
          & ( image(nat,nat,F5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),aa(nat,nat,F5,I3)) ) )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) )
              <=> ( aa(nat,nat,F5,I3) = aa(nat,nat,F5,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_7074_lenlex__append2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Us: list(A),Xs: list(A),Ys: list(A)] :
      ( irrefl(A,R)
     => ( 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys)),lenlex(A,R))
      <=> 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),lenlex(A,R)) ) ) ).

% lenlex_append2
tff(fact_7075_remdups__adj__Nil__iff,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( remdups_adj(A,Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% remdups_adj_Nil_iff
tff(fact_7076_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_7077_hd__remdups__adj,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),A,hd(A),remdups_adj(A,Xs)) = aa(list(A),A,hd(A),Xs) ).

% hd_remdups_adj
tff(fact_7078_lexord__same__pref__if__irrefl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( irrefl(A,R2)
     => ( 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)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lexord(A,R2))
      <=> 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),Zs),lexord(A,R2)) ) ) ).

% lexord_same_pref_if_irrefl
tff(fact_7079_rtranclp_Omono,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] : order_mono(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_ace(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),R2)) ).

% rtranclp.mono
tff(fact_7080_tranclp_Omono,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] : order_mono(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aTP_Lamp_acf(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),R2)) ).

% tranclp.mono
tff(fact_7081_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_7082_remdups__adj_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A] : remdups_adj(A,aa(list(A),list(A),cons(A,Xb),nil(A))) = aa(list(A),list(A),cons(A,Xb),nil(A)) ).

% remdups_adj.simps(2)
tff(fact_7083_remdups__adj_Oelims,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A)] :
      ( ( remdups_adj(A,Xb) = Y )
     => ( ( ( Xb = nil(A) )
         => ( Y != nil(A) ) )
       => ( ! [X4: A] :
              ( ( Xb = aa(list(A),list(A),cons(A,X4),nil(A)) )
             => ( Y != aa(list(A),list(A),cons(A,X4),nil(A)) ) )
         => ~ ! [X4: A,Y3: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Xs2)) )
               => ( Y != $ite(X4 = Y3,remdups_adj(A,aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(A),list(A),cons(A,X4),remdups_adj(A,aa(list(A),list(A),cons(A,Y3),Xs2)))) ) ) ) ) ) ).

% remdups_adj.elims
tff(fact_7084_remdups__adj_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups_adj(A,nil(A)) = nil(A) ).

% remdups_adj.simps(1)
tff(fact_7085_lexord__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
     => irrefl(list(A),lexord(A,R)) ) ).

% lexord_irrefl
tff(fact_7086_irrefl__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
     => irrefl(list(A),lex(A,R2)) ) ).

% irrefl_lex
tff(fact_7087_remdups__adj_Osimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A)] :
      remdups_adj(A,aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),cons(A,Y),Xs))) = $ite(Xb = Y,remdups_adj(A,aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(A),list(A),cons(A,Xb),remdups_adj(A,aa(list(A),list(A),cons(A,Y),Xs)))) ).

% remdups_adj.simps(3)
tff(fact_7088_remdups__adj__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups_adj(A,Xs) = Xs ) ) ).

% remdups_adj_distinct
tff(fact_7089_irrefl__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
    <=> ! [A6: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),A6),R2) ) ).

% irrefl_def
tff(fact_7090_irreflI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A4: A] : ~ 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)
     => irrefl(A,R) ) ).

% irreflI
tff(fact_7091_remdups__adj__append__two,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Y: A] :
      remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),cons(A,Y),nil(A))))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),nil(A))))),
        $ite(Xb = Y,nil(A),aa(list(A),list(A),cons(A,Y),nil(A)))) ).

% remdups_adj_append_two
tff(fact_7092_finite_Omono,axiom,
    ! [A: $tType] : order_mono(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_acg(fun(set(A),$o),fun(set(A),$o))) ).

% finite.mono
tff(fact_7093_lexl__not__refl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xb: list(A)] :
      ( irrefl(A,R2)
     => ~ 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),Xb),lex(A,R2)) ) ).

% lexl_not_refl
tff(fact_7094_remdups__adj__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),cons(A,Xb),nil(A)),aTP_Lamp_ach(A,fun(A,fun(list(A),list(A))),Xb)),remdups_adj(A,Xs)) ).

% remdups_adj_Cons
tff(fact_7095_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_aci(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_7096_remdups__adj__adjacent,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I)) ) ) ).

% remdups_adj_adjacent
tff(fact_7097_remdups__adj__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      remdups_adj(A,replicate(A,Nb,Xb)) = $ite(Nb = zero_zero(nat),nil(A),aa(list(A),list(A),cons(A,Xb),nil(A))) ).

% remdups_adj_replicate
tff(fact_7098_remdups__adj__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ( remdups_adj(A,Xs) = aa(list(A),list(A),cons(A,Xb),nil(A)) )
     => ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xb) ) ) ).

% remdups_adj_singleton
tff(fact_7099_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_acj(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ).

% lexordp.mono
tff(fact_7100_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_7101_remdups__adj__singleton__iff,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Xs != nil(A) )
        & ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),aa(list(A),A,hd(A),Xs)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_7102_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_qs(nat,fun(real,real),Nb),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_7103_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite(product_prod(A,A)),R)
     => ( transitive_rtrancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_abq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abv(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_7104_listrel__rtrancl__refl,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A))] : 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),listrel(A,A,transitive_rtrancl(A,R2))) ).

% listrel_rtrancl_refl
tff(fact_7105_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_7106_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_ack(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_7107_inj__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),top_top(set(product_prod(A,B))))
    <=> inj_on(A,C,F3,top_top(set(A))) ) ).

% inj_apfst
tff(fact_7108_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] :
      ( 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),F3),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,F3,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_7109_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A2: A,A5: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),insert(A,A2),A5))
    <=> ( inj_on(A,B,F3,A5)
        & ~ member(B,aa(A,B,F3,A2),image(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_7110_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: A,Xb: A,B2: A,F3: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),B2)
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
             => ( inj_on(A,B,F3,set_or1337092689740270186AtMost(A,A2,B2))
               => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),aa(A,B,F3,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,B2)) )
                  | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,B2)),aa(A,B,F3,Xb))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Xb)),aa(A,B,F3,A2)) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_7111_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C5: set(A),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),C5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
         => ( image(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),image(A,B,F3,A5)),image(A,B,F3,B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_7112_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),image(B,A,F3,A5))),aa(set(B),nat,finite_card(B),A5))
     => ~ inj_on(B,A,F3,A5) ) ).

% pigeonhole
tff(fact_7113_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
      ( 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))
     => ( 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,R2)))
       => 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)),aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(A),list(A),cons(A,Y),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_7114_listrel__rtrancl__trans,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Zs: list(A)] :
      ( 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),listrel(A,A,transitive_rtrancl(A,R2)))
     => ( 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),Zs),listrel(A,A,transitive_rtrancl(A,R2)))
       => 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),listrel(A,A,transitive_rtrancl(A,R2))) ) ) ).

% listrel_rtrancl_trans
tff(fact_7115_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$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),R2))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay)
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( 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),B4)),transitive_rtrancl(product_prod(A,B),R2))
             => ( 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),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R2)
               => ( aa(B,$o,aa(A,fun(B,$o),P,A4),B4)
                 => 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_7116_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xaa: A,Xb: B,Za: A,Zb: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( 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),R2))
     => ( ( 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,B4: B] :
              ( 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),B4)),R2)
             => ~ 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),B4)),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),R2)) ) ) ) ).

% converse_rtranclE2
tff(fact_7117_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,$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),R2))
     => ( aa(B,$o,aa(A,fun(B,$o),P,Bx),By)
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( 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),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R2)
             => ( 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),R2))
               => ( aa(B,$o,aa(A,fun(B,$o),P,Aa2),Ba)
                 => aa(B,$o,aa(A,fun(B,$o),P,A4),B4) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P,Ax),Ay) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_7118_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( 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,R2))
     => ( ( A22 != A1 )
       => ~ ! [B4: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4),transitive_rtrancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22),R2) ) ) ) ).

% rtrancl.cases
tff(fact_7119_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R2: set(product_prod(A,A))] :
      ( 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,R2))
    <=> ( ? [A6: A] :
            ( ( A1 = A6 )
            & ( A22 = A6 ) )
        | ? [A6: A,B6: A,C4: A] :
            ( ( A1 = A6 )
            & ( A22 = C4 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A6),B6),transitive_rtrancl(A,R2))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B6),C4),R2) ) ) ) ).

% rtrancl.simps
tff(fact_7120_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A2: A,R2: set(product_prod(A,A))] : 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,R2)) ).

% rtrancl.rtrancl_refl
tff(fact_7121_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( 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))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R2)
       => 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_rtrancl(A,R2)) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_7122_rtranclE,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( 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 )
       => ~ ! [Y3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y3),transitive_rtrancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),B2),R2) ) ) ) ).

% rtranclE
tff(fact_7123_rtrancl__trans,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),Z: A] :
      ( 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))
     => ( 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,R2))
       => 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,R2)) ) ) ).

% rtrancl_trans
tff(fact_7124_rtrancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$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))
     => ( aa(A,$o,P,A2)
       => ( ! [Y3: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y3),transitive_rtrancl(A,R2))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R2)
               => ( aa(A,$o,P,Y3)
                 => aa(A,$o,P,Z3) ) ) )
         => aa(A,$o,P,B2) ) ) ) ).

% rtrancl_induct
tff(fact_7125_converse__rtranclE,axiom,
    ! [A: $tType,Xb: A,Z: A,R2: set(product_prod(A,A))] :
      ( 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,R2))
     => ( ( Xb != Z )
       => ~ ! [Y3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y3),R2)
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z),transitive_rtrancl(A,R2)) ) ) ) ).

% converse_rtranclE
tff(fact_7126_converse__rtrancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P: fun(A,$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))
     => ( aa(A,$o,P,B2)
       => ( ! [Y3: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R2)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2),transitive_rtrancl(A,R2))
               => ( aa(A,$o,P,Z3)
                 => aa(A,$o,P,Y3) ) ) )
         => aa(A,$o,P,A2) ) ) ) ).

% converse_rtrancl_induct
tff(fact_7127_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( 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)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),transitive_rtrancl(A,R2))
       => 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_rtrancl(A,R2)) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_7128_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : listrel(A,A,transitive_rtrancl(A,R2)) = transitive_rtrancl(list(A),listrel1(A,R2)) ).

% listrel_rtrancl_eq_rtrancl_listrel1
tff(fact_7129_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A5: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),A5) ) ).

% inj_on_add
tff(fact_7130_inj__on__diff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) ) ).

% inj_on_diff
tff(fact_7131_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A5: set(A)] : inj_on(A,A,aTP_Lamp_acl(A,fun(A,A),A2),A5) ) ).

% inj_on_add'
tff(fact_7132_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,A5: set(A)] :
          ( ( A2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),A5) ) ) ).

% inj_on_mult
tff(fact_7133_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_ms(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_7134_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : inj_on(A,A,aTP_Lamp_of(A,fun(A,A),A2),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_7135_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_7136_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F3: fun(A,B)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
             => ( aa(A,B,F3,X4) != aa(A,B,F3,Y3) ) )
         => inj_on(A,B,F3,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_7137_option_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => inj_on(option(A),option(B),map_option(A,B,F3),top_top(set(option(A)))) ) ).

% option.inj_map
tff(fact_7138_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,transitive_rtrancl(A,R2))),transitive_rtrancl(list(A),listrel1(A,R2))) ).

% listrel1_rtrancl_subset_rtrancl_listrel1
tff(fact_7139_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X4: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X4),Y3)
             => ( member(A,X4,A5)
               => ( member(A,Y3,A5)
                 => ( aa(A,B,F3,X4) != aa(A,B,F3,Y3) ) ) ) )
         => ( ! [X4: A,Y3: A] :
                ( member(A,X4,A5)
               => ( member(A,Y3,A5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),Y3)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X4) ) ) )
           => inj_on(A,B,F3,A5) ) ) ) ).

% linorder_inj_onI
tff(fact_7140_image__set__diff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( image(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),image(A,B,F3,A5)),image(A,B,F3,B5)) ) ) ).

% image_set_diff
tff(fact_7141_inj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),Nb: nat] :
      ( inj_on(A,A,F3,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),F3),top_top(set(A))) ) ).

% inj_fn
tff(fact_7142_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( 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))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),transitive_rtrancl(A,R2))
       => 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,R2)) ) ) ).

% trancl_rtrancl_trancl
tff(fact_7143_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,Xb: A,Y: A,R2: set(product_prod(A,A)),Z: A] :
      ( 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))
     => ( 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,R2))
       => 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,R2)) ) ) ).

% rtrancl_trancl_trancl
tff(fact_7144_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( 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)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),transitive_rtrancl(A,R2))
       => 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,R2)) ) ) ).

% rtrancl_into_trancl2
tff(fact_7145_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( 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))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R2)
       => 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,R2)) ) ) ).

% rtrancl_into_trancl1
tff(fact_7146_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( 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))
    <=> ( ( Xb = Y )
        | ( ( Xb != Y )
          & 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)) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_7147_trancl__into__rtrancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( 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))
     => 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)) ) ).

% trancl_into_rtrancl
tff(fact_7148_tranclD2,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( 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))
     => ? [Z3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z3),transitive_rtrancl(A,R))
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y),R) ) ) ).

% tranclD2
tff(fact_7149_rtranclD,axiom,
    ! [A: $tType,A2: A,B2: A,R: set(product_prod(A,A))] :
      ( 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 )
        | ( ( A2 != B2 )
          & 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)) ) ) ) ).

% rtranclD
tff(fact_7150_tranclD,axiom,
    ! [A: $tType,Xb: A,Y: A,R: set(product_prod(A,A))] :
      ( 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))
     => ? [Z3: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Z3),R)
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y),transitive_rtrancl(A,R)) ) ) ).

% tranclD
tff(fact_7151_rtrancl__listrel1__ConsI1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Xb: A] :
      ( 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,R2)))
     => 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)),aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(A),list(A),cons(A,Xb),Ys)),transitive_rtrancl(list(A),listrel1(A,R2))) ) ).

% rtrancl_listrel1_ConsI1
tff(fact_7152_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( 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,R2)))
     => ( aa(list(A),nat,size_size(list(A)),Xb) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_7153_listrel__reflcl__if__listrel1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( 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,R2))
     => 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),listrel(A,A,transitive_rtrancl(A,R2))) ) ).

% listrel_reflcl_if_listrel1
tff(fact_7154_rtrancl__listrel1__if__listrel,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( 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),listrel(A,A,R2))
     => 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,R2))) ) ).

% rtrancl_listrel1_if_listrel
tff(fact_7155_listrel__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R2)),transitive_rtrancl(list(A),listrel1(A,R2))) ).

% listrel_subset_rtrancl_listrel1
tff(fact_7156_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(B),$o,finite_finite(B),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5))
         => ? [F2: fun(A,B)] :
              ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,F2,A5)),B5)
              & inj_on(A,B,F2,A5) ) ) ) ) ).

% card_le_inj
tff(fact_7157_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(B)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,F3,A5)),B5)
       => ( aa(set(B),$o,finite_finite(B),B5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)) ) ) ) ).

% card_inj_on_le
tff(fact_7158_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(B),$o,finite_finite(B),B5)
       => ( ? [F5: fun(A,B)] :
              ( inj_on(A,B,F5,A5)
              & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),image(A,B,F5,A5)),B5) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_7159_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),set_ord_greaterThan(real,zero_zero(real))) ) ).

% log_inj
tff(fact_7160_funpow__inj__finite,axiom,
    ! [A: $tType,P2: fun(A,A),Xb: A] :
      ( inj_on(A,A,P2,top_top(set(A)))
     => ( aa(set(A),$o,finite_finite(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_acm(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_7161_ran__map__of__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( ran(A,B,map_of(A,B,zip(A,B,Xs,Ys))) = aa(list(B),set(B),set2(B),Ys) ) ) ) ).

% ran_map_of_zip
tff(fact_7162_shuffles_Oelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Y: set(list(A))] :
      ( ( shuffles(A,Xb,Xaa) = Y )
     => ( ( ( Xb = nil(A) )
         => ( Y != aa(set(list(A)),set(list(A)),insert(list(A),Xaa),bot_bot(set(list(A)))) ) )
       => ( ( ( Xaa = nil(A) )
           => ( Y != aa(set(list(A)),set(list(A)),insert(list(A),Xb),bot_bot(set(list(A)))) ) )
         => ~ ! [X4: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),cons(A,X4),Xs2) )
               => ! [Y3: A,Ys3: list(A)] :
                    ( ( Xaa = aa(list(A),list(A),cons(A,Y3),Ys3) )
                   => ( Y != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),image(list(A),list(A),cons(A,X4),shuffles(A,Xs2,aa(list(A),list(A),cons(A,Y3),Ys3)))),image(list(A),list(A),cons(A,Y3),shuffles(A,aa(list(A),list(A),cons(A,X4),Xs2),Ys3))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_7163_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% sup.bounded_iff
tff(fact_7164_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_7165_Un__Diff__cancel,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) ).

% Un_Diff_cancel
tff(fact_7166_Un__Diff__cancel2,axiom,
    ! [A: $tType,B5: set(A),A5: 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)),B5),A5)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A5) ).

% Un_Diff_cancel2
tff(fact_7167_set__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(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_7168_Compl__Diff__eq,axiom,
    ! [A: $tType,A5: set(A),B5: 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)),A5),B5)) = 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)),A5)),B5) ).

% Compl_Diff_eq
tff(fact_7169_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_acn(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_7170_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_7171_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F3: fun(A,B),A5: A,B5: A] :
          ( order_mono(A,B,F3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,A5)),aa(A,B,F3,B5))),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B5))) ) ) ).

% mono_sup
tff(fact_7172_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A5: 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_mb(A,fun(B,product_prod(B,A)))),A5) ).

% swap_inj_on
tff(fact_7173_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X6: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_aco(fun(A,B),fun(A,product_prod(A,B)),F3),X6) ).

% inj_on_convol_ident
tff(fact_7174_inj__Some,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,option(A),some(A),A5) ).

% inj_Some
tff(fact_7175_inj__on__Cons1,axiom,
    ! [A: $tType,Xb: A,A5: set(list(A))] : inj_on(list(A),list(A),cons(A,Xb),A5) ).

% inj_on_Cons1
tff(fact_7176_inj__list__encode,axiom,
    ! [A5: set(list(nat))] : inj_on(list(nat),nat,nat_list_encode,A5) ).

% inj_list_encode
tff(fact_7177_inj__Suc,axiom,
    ! [N3: set(nat)] : inj_on(nat,nat,suc,N3) ).

% inj_Suc
tff(fact_7178_inj__prod__encode,axiom,
    ! [A5: set(product_prod(nat,nat))] : inj_on(product_prod(nat,nat),nat,nat_prod_encode,A5) ).

% inj_prod_encode
tff(fact_7179_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_7180_inj__split__Cons,axiom,
    ! [A: $tType,X6: 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_acp(list(A),fun(A,list(A)))),X6) ).

% inj_split_Cons
tff(fact_7181_inj__on__diff__nat,axiom,
    ! [N3: set(nat),K2: nat] :
      ( ! [N: nat] :
          ( member(nat,N,N3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N) )
     => inj_on(nat,nat,aTP_Lamp_nz(nat,fun(nat,nat),K2),N3) ) ).

% inj_on_diff_nat
tff(fact_7182_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),finite_finite(nat))) ).

% inj_on_set_encode
tff(fact_7183_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_7184_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_7185_Diff__subset__conv,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),C5)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C5)) ) ).

% Diff_subset_conv
tff(fact_7186_Diff__partition,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) = B5 ) ) ).

% Diff_partition
tff(fact_7187_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_7188_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_7189_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_7190_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_7191_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_7192_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_7193_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_7194_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_7195_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A2: A,D3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_7196_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D3)
           => aa(A,$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),C3),D3)) ) ) ) ).

% sup_mono
tff(fact_7197_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_7198_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_7199_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_7200_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_7201_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F3: fun(A,fun(A,A)),Xb: A,Y: A] :
          ( ! [X4: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,aa(A,fun(A,A),F3,X4),Y3))
         => ( ! [X4: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F3,X4),Y3))
           => ( ! [X4: A,Y3: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),X4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3)),X4) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),F3,Xb),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_7202_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_7203_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_7204_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_7205_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_7206_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% sup.boundedE
tff(fact_7207_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2) ) ) ) ).

% sup.boundedI
tff(fact_7208_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_7209_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_7210_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_7211_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_7212_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_7213_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI1
tff(fact_7214_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI2
tff(fact_7215_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_7216_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_7217_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_7218_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_7219_Un__Diff__Int,axiom,
    ! [A: $tType,A5: set(A),B5: 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)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = A5 ).

% Un_Diff_Int
tff(fact_7220_Int__Diff__Un,axiom,
    ! [A: $tType,A5: set(A),B5: 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)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = A5 ).

% Int_Diff_Un
tff(fact_7221_Diff__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),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)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),C5)) ).

% Diff_Int
tff(fact_7222_Diff__Un,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),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)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),C5)) ).

% Diff_Un
tff(fact_7223_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_7224_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_7225_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_7226_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_7227_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% sup.strict_boundedE
tff(fact_7228_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_7229_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI1
tff(fact_7230_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI2
tff(fact_7231_Un__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: 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)),A5),B5)),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)),A5),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C5)) ).

% Un_Diff
tff(fact_7232_ranI,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),Mb,A2) = aa(A,option(A),some(A),B2) )
     => member(A,B2,ran(B,A,Mb)) ) ).

% ranI
tff(fact_7233_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys))
     => ( aa(list(A),set(A),set2(A),Zs) = 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_7234_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P2: A,Q3: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q3),R2))
        <=> aa(A,$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),Q3))),R2) ) ) ).

% sup_neg_inf
tff(fact_7235_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_7236_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_7237_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] : aa(A,$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),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))),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)),A5),B5))) ) ).

% less_eq_Inf_inter
tff(fact_7238_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_7239_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)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_7240_card__Un__le,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(nat,$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)),A5),B5))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))) ).

% card_Un_le
tff(fact_7241_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_7242_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)),set_ord_atLeast(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_7243_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)),set_ord_atMost(A,L)),set_or3652927894154168847AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_7244_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)),set_ord_greaterThan(A,U)) = set_ord_greaterThan(A,L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_7245_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ? [N: nat,F2: fun(nat,A)] :
          ( ( A5 = image(nat,A,F2,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),N))) )
          & inj_on(nat,A,F2,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_7246_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ? [F2: fun(A,nat),N: nat] :
          ( ( image(A,nat,F2,A5) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),N)) )
          & inj_on(A,nat,F2,A5) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_7247_ran__def,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A))] : ran(B,A,Mb) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_acq(fun(B,option(A)),fun(A,$o),Mb)) ).

% ran_def
tff(fact_7248_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A)] : shuffles(A,aa(list(A),list(A),cons(A,Xb),Xs),aa(list(A),list(A),cons(A,Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),image(list(A),list(A),cons(A,Xb),shuffles(A,Xs,aa(list(A),list(A),cons(A,Y),Ys)))),image(list(A),list(A),cons(A,Y),shuffles(A,aa(list(A),list(A),cons(A,Xb),Xs),Ys))) ).

% shuffles.simps(3)
tff(fact_7249_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I6: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X4: nat] :
            ( member(nat,X4,I6)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X4),aa(list(A),nat,size_size(list(A)),Xs)) )
       => inj_on(nat,A,nth(A,Xs),I6) ) ) ).

% inj_on_nth
tff(fact_7250_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_7251_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),B5)) ) ) ) ) ).

% sum.union_inter
tff(fact_7252_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) = 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),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ).

% prod.union_inter
tff(fact_7253_card__Un__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(A),$o,finite_finite(A),B5)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) = 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)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ).

% card_Un_Int
tff(fact_7254_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_7255_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_7256_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)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_7257_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_7258_inj__on__funpow__least,axiom,
    ! [A: $tType,Nb: nat,F3: fun(A,A),S2: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),F3),S2) = S2 )
     => ( ! [M: nat] :
            ( aa(nat,$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),F3),S2) != S2 ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_acr(fun(A,A),fun(A,fun(nat,A)),F3),S2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ) ).

% inj_on_funpow_least
tff(fact_7259_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)),set_ord_atMost(A,L)),set_or5935395276787703475ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_7260_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)),set_ord_greaterThan(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_7261_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_7262_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: A,B5: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),aa(set(A),A,complete_Sup_Sup(A),image(nat,A,aTP_Lamp_py(A,fun(nat,A),B5),aa(fun(nat,$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),A5),B5) ) ).

% SUP_nat_binary
tff(fact_7263_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)),set_ord_atLeast(A,U)) = set_ord_greaterThan(A,L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_7264_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),lattic7752659483105999362nf_fin(A,A5)),lattic7752659483105999362nf_fin(A,B5)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_acs(set(A),fun(set(A),fun(A,$o)),A5),B5))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_7265_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xb),lattic7752659483105999362nf_fin(A,A5)) = lattic7752659483105999362nf_fin(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_act(set(A),fun(A,fun(A,$o)),A5),Xb))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_7266_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_acu(A,fun(A,$o)),aTP_Lamp_acv(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_7267_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
                 => ( aa(A,B,G3,X4) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),B5)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_7268_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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,F3),A5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),B5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ).

% sum_Un
tff(fact_7269_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),A5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),B5)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_7270_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
                 => ( aa(A,B,G3,X4) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_7271_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_7272_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_7273_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ).

% sum_Un2
tff(fact_7274_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ).

% sum.union_diff2
tff(fact_7275_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ).

% prod.union_diff2
tff(fact_7276_card__Un__disjoint,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(A),$o,finite_finite(A),B5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = 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)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_7277_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
    <=> ( inj_on(A,B,F3,A5)
        & inj_on(A,B,F3,B5)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),image(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),image(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_7278_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_7279_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_7280_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_7281_sum__Un__nat,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(A),$o,finite_finite(A),B5)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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,F3),A5)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),B5))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ).

% sum_Un_nat
tff(fact_7282_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_7283_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_7284_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( aa(set(A),$o,finite_finite(A),B5)
           => ( ! [X4: A] :
                  ( member(A,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
                 => ( aa(A,B,F3,X4) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ) ).

% prod_Un
tff(fact_7285_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_7286_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)),image(nat,set(A),M6,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)),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_7287_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] : lattic5882676163264333800up_fin(A,A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_acw(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Sup_fin.eq_fold'
tff(fact_7288_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),I: B,B5: set(A),J4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),fun_upd(B,set(A),A5,I,B5),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)),image(B,set(A),A5,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,I),bot_bot(set(B))))))),
        $ite(member(B,I,J4),B5,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_7289_rtrancl__Un__separator__converseE,axiom,
    ! [A: $tType,A2: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( 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)))
     => ( ! [X4: A,Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),B2),transitive_rtrancl(A,P))
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),Q)
             => ( Y3 = X4 ) ) )
       => 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_7290_rtrancl__Un__separatorE,axiom,
    ! [A: $tType,A2: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
      ( 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)))
     => ( ! [X4: A,Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),X4),transitive_rtrancl(A,P))
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),Q)
             => ( X4 = Y3 ) ) )
       => 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_7291_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_7292_sup__enat__def,axiom,
    sup_sup(extended_enat) = ord_max(extended_enat) ).

% sup_enat_def
tff(fact_7293_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B)),X: A,Xa: B] :
      ( aa(B,$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_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),S3)),X),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),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))),R),S3)) ) ).

% sup_Un_eq2
tff(fact_7294_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),lattic5882676163264333800up_fin(A,A5)) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_7295_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_7296_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A5)),Xb)
             => ! [A8: A] :
                  ( member(A,A8,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A8),Xb) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_7297_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => 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,A5)),Xb) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_7298_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A5)),Xb)
            <=> ! [X3: A] :
                  ( member(A,X3,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xb) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_7299_rtrancl__insert,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R2)),aa(fun(product_prod(A,A),$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)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_acx(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A2),B2),R2)))) ).

% rtrancl_insert
tff(fact_7300_trancl__insert2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$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)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_acy(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A2),B2),R2)))) ).

% trancl_insert2
tff(fact_7301_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite(A),A5)
         => ( lattic5882676163264333800up_fin(A,A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_7302_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic5882676163264333800up_fin(A,A5)),lattic5882676163264333800up_fin(A,B5)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_7303_trancl__insert,axiom,
    ! [A: $tType,Y: A,Xb: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xb)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$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)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_acx(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Y),Xb),R2)))) ).

% trancl_insert
tff(fact_7304_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic7752659483105999362nf_fin(A,A5)),lattic5882676163264333800up_fin(A,A5)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_7305_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),lattic5882676163264333800up_fin(A,A5)),lattic5882676163264333800up_fin(A,B5)) = lattic5882676163264333800up_fin(A,aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_acz(set(A),fun(set(A),fun(A,$o)),A5),B5))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_7306_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xb),lattic5882676163264333800up_fin(A,A5)) = lattic5882676163264333800up_fin(A,aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ada(set(A),fun(A,fun(A,$o)),A5),Xb))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_7307_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xb: B,Y: A,A5: set(B)] :
      image(B,A,fun_upd(B,A,F3,Xb,Y),A5) = $ite(member(B,Xb,A5),aa(set(A),set(A),insert(A,Y),image(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),insert(B,Xb),bot_bot(set(B)))))),image(B,A,F3,A5)) ).

% fun_upd_image
tff(fact_7308_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( lattic5882676163264333800up_fin(A,aa(set(A),set(A),insert(A,Xb),A5)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_7309_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xb: A] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( member(A,Xb,A5)
           => ( lattic5882676163264333800up_fin(A,A5) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),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)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_7310_Pow__set_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      pow2(A,aa(list(A),set(A),set2(A),aa(list(A),list(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),image(set(A),set(A),insert(A,Xb),a3)) ) ).

% Pow_set(2)
tff(fact_7311_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B5: set(A),Xb: B,A5: set(product_prod(B,A))] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => ( 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))),image(A,set(product_prod(B,A)),aTP_Lamp_adb(B,fun(A,set(product_prod(B,A))),Xb),B5))),A5) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_adc(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Xb),A5,B5) ) ) ).

% fold_union_pair
tff(fact_7312_ran__distinct,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
     => ( ran(A,B,map_of(A,B,Al)) = image(product_prod(A,B),B,product_snd(A,B),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al)) ) ) ).

% ran_distinct
tff(fact_7313_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A),Y: A] :
          ( ( min_list(A,Xb) = Y )
         => ( ! [X4: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),cons(A,X4),Xs2) )
               => ( Y != aa(list(A),A,case_list(A,A,X4,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_abo(A,fun(list(A),fun(A,fun(list(A),A))),X4),Xs2)),Xs2) ) )
           => ~ ( ( Xb = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_7314_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,Xb: A,X: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_ph(A,option(B)),Xb,none(B)),X) = none(B) ).

% empty_upd_none
tff(fact_7315_map__ident,axiom,
    ! [A: $tType,X: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_nt(A,A)),X) = X ).

% map_ident
tff(fact_7316_list_Omap__disc__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A2: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),A2) = nil(A) )
    <=> ( A2 = nil(B) ) ) ).

% list.map_disc_iff
tff(fact_7317_Nil__is__map__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] :
      ( ( nil(A) = aa(list(B),list(A),map(B,A,F3),Xs) )
    <=> ( Xs = nil(B) ) ) ).

% Nil_is_map_conv
tff(fact_7318_map__is__Nil__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = nil(A) )
    <=> ( Xs = nil(B) ) ) ).

% map_is_Nil_conv
tff(fact_7319_list_Omap__comp,axiom,
    ! [B: $tType,A: $tType,C: $tType,G3: fun(B,A),F3: fun(C,B),V: list(C)] : aa(list(B),list(A),map(B,A,G3),aa(list(C),list(B),map(C,B,F3),V)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,G3),F3)),V) ).

% list.map_comp
tff(fact_7320_List_Omap_Ocompositionality,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G3: fun(C,B),List: list(C)] : aa(list(B),list(A),map(B,A,F3),aa(list(C),list(B),map(C,B,G3),List)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F3),G3)),List) ).

% List.map.compositionality
tff(fact_7321_map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G3: fun(C,B),Xs: list(C)] : aa(list(B),list(A),map(B,A,F3),aa(list(C),list(B),map(C,B,G3),Xs)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F3),G3)),Xs) ).

% map_map
tff(fact_7322_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),G3: fun(B,A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,G3),Xs) )
    <=> ! [X3: B] :
          ( member(B,X3,aa(list(B),set(B),set2(B),Xs))
         => ( aa(B,A,F3,X3) = aa(B,A,G3,X3) ) ) ) ).

% map_eq_conv
tff(fact_7323_length__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_7324_map__append,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F3),Xs)),aa(list(B),list(A),map(B,A,F3),Ys)) ).

% map_append
tff(fact_7325_list_Omap__id0,axiom,
    ! [A: $tType] : map(A,A,id(A)) = id(list(A)) ).

% list.map_id0
tff(fact_7326_map__replicate,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Nb: nat,Xb: B] : aa(list(B),list(A),map(B,A,F3),replicate(B,Nb,Xb)) = replicate(A,Nb,aa(B,A,F3,Xb)) ).

% map_replicate
tff(fact_7327_image__map__upd,axiom,
    ! [B: $tType,A: $tType,Xb: A,A5: set(A),Mb: fun(A,option(B)),Y: B] :
      ( ~ member(A,Xb,A5)
     => ( image(A,option(B),fun_upd(A,option(B),Mb,Xb,aa(B,option(B),some(B),Y)),A5) = image(A,option(B),Mb,A5) ) ) ).

% image_map_upd
tff(fact_7328_list_Oset__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),V: list(B)] : aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F3),V)) = image(B,A,F3,aa(list(B),set(B),set2(B),V)) ).

% list.set_map
tff(fact_7329_map__snd__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(A),map(product_prod(nat,A),A,product_snd(nat,A)),enumerate(A,Nb,Xs)) = Xs ).

% map_snd_enumerate
tff(fact_7330_inj__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,F3),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_map_eq_map
tff(fact_7331_map__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y: A,Xs: list(A),F3: fun(A,B),V: B] :
      ( ~ member(A,Y,aa(list(A),set(A),set2(A),Xs))
     => ( aa(list(A),list(B),map(A,B,fun_upd(A,B,F3,Y,V)),Xs) = aa(list(A),list(B),map(A,B,F3),Xs) ) ) ).

% map_fun_upd
tff(fact_7332_map__option__o__map__upd,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),Mb: fun(A,option(C)),A2: A,B2: C] : aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),fun_upd(A,option(C),Mb,A2,aa(C,option(C),some(C),B2))) = fun_upd(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),Mb),A2,aa(B,option(B),some(B),aa(C,B,F3,B2))) ).

% map_option_o_map_upd
tff(fact_7333_List_Omap_Ocomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),G3: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),comp(list(C),list(B),list(A),map(C,B,F3)),map(A,C,G3)) = map(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,F3),G3)) ).

% List.map.comp
tff(fact_7334_map__comp__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),G3: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),comp(list(C),list(B),list(A),map(C,B,F3)),map(A,C,G3)) = map(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,F3),G3)) ).

% map_comp_map
tff(fact_7335_size__list__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F3),aa(list(B),list(A),map(B,A,G3),Xs)) = aa(list(B),nat,size_list(B,aa(fun(B,A),fun(B,nat),comp(A,nat,B,F3),G3)),Xs) ).

% size_list_map
tff(fact_7336_nth__map,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),F3: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F3),Xs)),Nb) = aa(A,B,F3,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% nth_map
tff(fact_7337_map__fst__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),zip(A,B,Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_7338_map__snd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),zip(A,B,Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_7339_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),Mb,A2) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),Mb,A2,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),insert(A,B2),ran(B,A,Mb)) ) ) ).

% ran_map_upd
tff(fact_7340_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_add(fun(B,A),fun(B,list(A)),F3)),Xs)) = aa(list(B),list(A),map(B,A,F3),Xs) ).

% concat_map_singleton
tff(fact_7341_inj__mapI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => inj_on(list(A),list(B),map(A,B,F3),top_top(set(list(A)))) ) ).

% inj_mapI
tff(fact_7342_inj__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F3),top_top(set(list(A))))
    <=> inj_on(A,B,F3,top_top(set(A))) ) ).

% inj_map
tff(fact_7343_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),Xb: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))
       => ( aa(A,option(B),map_of(A,B,Xys),Xb) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_is_SomeI
tff(fact_7344_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,Xb: A] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),Xb) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).

% Some_eq_map_of_iff
tff(fact_7345_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Xb: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(A,option(B),map_of(A,B,Xys),Xb) = aa(B,option(B),some(B),Y) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).

% map_of_eq_Some_iff
tff(fact_7346_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F3,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)))
     => ( ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,F3),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_on_map_eq_map
tff(fact_7347_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
     => ( inj_on(B,A,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(list(B),set(B),set2(B),Xs)),aa(list(B),set(B),set2(B),Ys)))
       => ( Xs = Ys ) ) ) ).

% map_inj_on
tff(fact_7348_map__injective,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
     => ( inj_on(B,A,F3,top_top(set(B)))
       => ( Xs = Ys ) ) ) ).

% map_injective
tff(fact_7349_distinct__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
    <=> ( distinct(B,Xs)
        & inj_on(B,A,F3,aa(list(B),set(B),set2(B),Xs)) ) ) ).

% distinct_map
tff(fact_7350_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_ade(fun(C,A),fun(C,fun(B,product_prod(A,B))),F3))),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_7351_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F3: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F3),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_adf(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3))),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_7352_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(product_prod(B,C),A),G3: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,aa(list(D),list(B),map(D,B,G3),Xs),Ys)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_adg(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F3),G3))),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_7353_zip__map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,F3: fun(C,A),Xs: list(C),G3: fun(D,B),Ys: list(D)] : zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),aa(list(D),list(B),map(D,B,G3),Ys)) = aa(list(product_prod(C,D)),list(product_prod(A,B)),map(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_adh(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3))),zip(C,D,Xs,Ys)) ).

% zip_map_map
tff(fact_7354_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(product_prod(B,C),A),Xs: list(B),G3: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G3),Ys))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_adi(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F3),G3))),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_7355_map__upd__Some__unfold,axiom,
    ! [B: $tType,A: $tType,Mb: fun(B,option(A)),A2: B,B2: A,Xb: B,Y: A] :
      ( ( aa(B,option(A),fun_upd(B,option(A),Mb,A2,aa(A,option(A),some(A),B2)),Xb) = aa(A,option(A),some(A),Y) )
    <=> ( ( ( Xb = A2 )
          & ( B2 = Y ) )
        | ( ( Xb != A2 )
          & ( aa(B,option(A),Mb,Xb) = aa(A,option(A),some(A),Y) ) ) ) ) ).

% map_upd_Some_unfold
tff(fact_7356_map__upd__triv,axiom,
    ! [A: $tType,B: $tType,Ta: fun(B,option(A)),K2: B,Xb: A] :
      ( ( aa(B,option(A),Ta,K2) = aa(A,option(A),some(A),Xb) )
     => ( fun_upd(B,option(A),Ta,K2,aa(A,option(A),some(A),Xb)) = Ta ) ) ).

% map_upd_triv
tff(fact_7357_map__upd__eqD1,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),A2: A,Xb: B,Nb: fun(A,option(B)),Y: B] :
      ( ( fun_upd(A,option(B),Mb,A2,aa(B,option(B),some(B),Xb)) = fun_upd(A,option(B),Nb,A2,aa(B,option(B),some(B),Y)) )
     => ( Xb = Y ) ) ).

% map_upd_eqD1
tff(fact_7358_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,Ta: fun(A,option(B)),K2: A,Xb: B] :
      ~ ! [X4: A] : aa(A,option(B),fun_upd(A,option(B),Ta,K2,aa(B,option(B),some(B),Xb)),X4) = none(B) ).

% map_upd_nonempty
tff(fact_7359_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F3: fun(D,B),Xs: list(D),G3: fun(D,C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H)),zip(B,C,aa(list(D),list(B),map(D,B,F3),Xs),aa(list(D),list(C),map(D,C,G3),Xs))) = aa(list(D),list(A),map(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_adj(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F3),G3)),Xs) ).

% map2_map_map
tff(fact_7360_remdups__adj__map__injective,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( remdups_adj(B,aa(list(A),list(B),map(A,B,F3),Xs)) = aa(list(A),list(B),map(A,B,F3),remdups_adj(A,Xs)) ) ) ).

% remdups_adj_map_injective
tff(fact_7361_zip__map__fst__snd,axiom,
    ! [B: $tType,A: $tType,Zs: list(product_prod(A,B))] : zip(A,B,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Zs),aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Zs)) = Zs ).

% zip_map_fst_snd
tff(fact_7362_map__of__eqI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)) = aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)) )
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
           => ( aa(A,option(B),map_of(A,B,Xs),X4) = aa(A,option(B),map_of(A,B,Ys),X4) ) )
       => ( map_of(A,B,Xs) = map_of(A,B,Ys) ) ) ) ).

% map_of_eqI
tff(fact_7363_image__set,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : image(B,A,F3,aa(list(B),set(B),set2(B),Xs)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F3),Xs)) ).

% image_set
tff(fact_7364_list_Omap__sel_I1_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F3: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F3),A2)) = aa(A,B,F3,aa(list(A),A,hd(A),A2)) ) ) ).

% list.map_sel(1)
tff(fact_7365_hd__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,B)] :
      ( ( Xs != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F3),Xs)) = aa(A,B,F3,aa(list(A),A,hd(A),Xs)) ) ) ).

% hd_map
tff(fact_7366_list_Osimps_I8_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] : aa(list(B),list(A),map(B,A,F3),nil(B)) = nil(A) ).

% list.simps(8)
tff(fact_7367_drop__map,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F3: fun(B,A),Xs: list(B)] : drop(A,Nb,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),drop(B,Nb,Xs)) ).

% drop_map
tff(fact_7368_remdups__map__remdups,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : remdups(A,aa(list(B),list(A),map(B,A,F3),remdups(B,Xs))) = remdups(A,aa(list(B),list(A),map(B,A,F3),Xs)) ).

% remdups_map_remdups
tff(fact_7369_take__map,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F3: fun(B,A),Xs: list(B)] : take(A,Nb,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),take(B,Nb,Xs)) ).

% take_map
tff(fact_7370_map__update,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),K2: nat,Y: B] : aa(list(B),list(A),map(B,A,F3),list_update(B,Xs,K2,Y)) = list_update(A,aa(list(B),list(A),map(B,A,F3),Xs),K2,aa(B,A,F3,Y)) ).

% map_update
tff(fact_7371_rotate1__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate1(A),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),rotate1(B),Xs)) ).

% rotate1_map
tff(fact_7372_map__concat,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(list(B))] : aa(list(B),list(A),map(B,A,F3),concat(B,Xs)) = concat(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F3)),Xs)) ).

% map_concat
tff(fact_7373_list_Omap__ident,axiom,
    ! [A: $tType,Ta: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_nt(A,A)),Ta) = Ta ).

% list.map_ident
tff(fact_7374_pair__list__eqI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs) = aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys) )
     => ( ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs) = aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_7375_nths__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),I6: set(nat)] : nths(A,aa(list(B),list(A),map(B,A,F3),Xs),I6) = aa(list(B),list(A),map(B,A,F3),nths(B,Xs,I6)) ).

% nths_map
tff(fact_7376_list_Omap__id,axiom,
    ! [A: $tType,Ta: list(A)] : aa(list(A),list(A),map(A,A,id(A)),Ta) = Ta ).

% list.map_id
tff(fact_7377_List_Omap_Oidentity,axiom,
    ! [A: $tType] : map(A,A,aTP_Lamp_nt(A,A)) = id(list(A)) ).

% List.map.identity
tff(fact_7378_enumerate__Suc__eq,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : enumerate(A,aa(nat,nat,suc,Nb),Xs) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),map(product_prod(nat,A),product_prod(nat,A),product_apfst(nat,nat,A,suc)),enumerate(A,Nb,Xs)) ).

% enumerate_Suc_eq
tff(fact_7379_list_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,nat),G3: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),nat),comp(list(B),nat,list(A),size_list(B,F3)),map(A,B,G3)) = size_list(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F3),G3)) ).

% list.size_gen_o_map
tff(fact_7380_list_Osimps_I9_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X21: B,X222: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),cons(B,X21),X222)) = aa(list(A),list(A),cons(A,aa(B,A,F3,X21)),aa(list(B),list(A),map(B,A,F3),X222)) ).

% list.simps(9)
tff(fact_7381_Cons__eq__map__D,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),F3: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),cons(A,Xb),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
     => ? [Z3: B,Zs2: list(B)] :
          ( ( Ys = aa(list(B),list(B),cons(B,Z3),Zs2) )
          & ( Xb = aa(B,A,F3,Z3) )
          & ( Xs = aa(list(B),list(A),map(B,A,F3),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_7382_map__eq__Cons__D,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(A),list(A),cons(A,Y),Ys) )
     => ? [Z3: B,Zs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),cons(B,Z3),Zs2) )
          & ( aa(B,A,F3,Z3) = Y )
          & ( aa(list(B),list(A),map(B,A,F3),Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_7383_Cons__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),F3: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),cons(A,Xb),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
    <=> ? [Z2: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),cons(B,Z2),Zs3) )
          & ( Xb = aa(B,A,F3,Z2) )
          & ( Xs = aa(list(B),list(A),map(B,A,F3),Zs3) ) ) ) ).

% Cons_eq_map_conv
tff(fact_7384_map__eq__Cons__conv,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(A),list(A),cons(A,Y),Ys) )
    <=> ? [Z2: B,Zs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),cons(B,Z2),Zs3) )
          & ( aa(B,A,F3,Z2) = Y )
          & ( aa(list(B),list(A),map(B,A,F3),Zs3) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_7385_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,A),Xs: list(B),G3: fun(C,A),Ys: list(C)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(C),list(A),map(C,A,G3),Ys) )
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_7386_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K2: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_lp(A,fun(B,A)),K2)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K2) ).

% map_replicate_const
tff(fact_7387_map__eq__append__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( Ys = aa(list(B),list(A),map(B,A,F3),Us2) )
          & ( Zs = aa(list(B),list(A),map(B,A,F3),Vs3) ) ) ) ).

% map_eq_append_conv
tff(fact_7388_append__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Ys: list(A),Zs: list(A),F3: fun(B,A),Xs: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(B),list(A),map(B,A,F3),Xs) )
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( Ys = aa(list(B),list(A),map(B,A,F3),Us2) )
          & ( Zs = aa(list(B),list(A),map(B,A,F3),Vs3) ) ) ) ).

% append_eq_map_conv
tff(fact_7389_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ys: list(B),F3: fun(A,B)] :
      ( ? [Xs3: list(A)] : Ys = aa(list(A),list(B),map(A,B,F3),Xs3)
    <=> ! [X3: B] :
          ( member(B,X3,aa(list(B),set(B),set2(B),Ys))
         => ? [Xa2: A] : X3 = aa(A,B,F3,Xa2) ) ) ).

% ex_map_conv
tff(fact_7390_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( Xs = Ys )
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,B,F3,X4) = aa(A,B,G3,X4) ) )
       => ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,G3),Ys) ) ) ) ).

% map_cong
tff(fact_7391_map__idI,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,A)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,A,F3,X4) = X4 ) )
     => ( aa(list(A),list(A),map(A,A,F3),Xs) = Xs ) ) ).

% map_idI
tff(fact_7392_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,B,F3,X4) = aa(A,B,G3,X4) ) )
     => ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,G3),Xs) ) ) ).

% map_ext
tff(fact_7393_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Xaa: list(A),F3: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z3: A,Za2: A] :
          ( member(A,Z3,aa(list(A),set(A),set2(A),Xb))
         => ( member(A,Za2,aa(list(A),set(A),set2(A),Xaa))
           => ( ( aa(A,B,F3,Z3) = aa(A,B,Fa,Za2) )
             => ( Z3 = Za2 ) ) ) )
     => ( ( aa(list(A),list(B),map(A,B,F3),Xb) = aa(list(A),list(B),map(A,B,Fa),Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_7394_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ! [Z3: A] :
          ( member(A,Z3,aa(list(A),set(A),set2(A),Xb))
         => ( aa(A,B,F3,Z3) = aa(A,B,G3,Z3) ) )
     => ( aa(list(A),list(B),map(A,B,F3),Xb) = aa(list(A),list(B),map(A,B,G3),Xb) ) ) ).

% list.map_cong0
tff(fact_7395_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Ya: list(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( Xb = Ya )
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,B,F3,Z3) = aa(A,B,G3,Z3) ) )
       => ( aa(list(A),list(B),map(A,B,F3),Xb) = aa(list(A),list(B),map(A,B,G3),Ya) ) ) ) ).

% list.map_cong
tff(fact_7396_map__removeAll__inj,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xb: A,Xs: list(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),removeAll(A,Xb),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F3,Xb)),aa(list(A),list(B),map(A,B,F3),Xs)) ) ) ).

% map_removeAll_inj
tff(fact_7397_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = case_option(A,A,undefined(A),aTP_Lamp_nt(A,A),Option) ).

% option.the_def
tff(fact_7398_hd__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),A,hd(A),List) = aa(list(A),A,case_list(A,A,undefined(A),aTP_Lamp_adk(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_7399_zip__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_adl(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs)) ).

% zip_assoc
tff(fact_7400_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_adn(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs))) ).

% zip_left_commute
tff(fact_7401_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),map(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_mc(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_7402_zip__eq__conv,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(product_prod(A,B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( zip(A,B,Xs,Ys) = Zs )
      <=> ( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Zs) = Xs )
          & ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Zs) = Ys ) ) ) ) ).

% zip_eq_conv
tff(fact_7403_distinct__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xb: B,Xs: list(B)] :
          ( distinct(A,aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),Xs)))
        <=> ( ~ member(A,aa(B,A,F3,Xb),image(B,A,F3,aa(list(B),set(B),set2(B),Xs)))
            & distinct(A,aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ) ).

% distinct_insort_key
tff(fact_7404_map__removeAll__inj__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xb: A,Xs: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),insert(A,Xb),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),removeAll(A,Xb),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F3,Xb)),aa(list(A),list(B),map(A,B,F3),Xs)) ) ) ).

% map_removeAll_inj_on
tff(fact_7405_eq__key__imp__eq__value,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K2: A,V1: B,V22: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V1),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V22),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
         => ( V1 = V22 ) ) ) ) ).

% eq_key_imp_eq_value
tff(fact_7406_map__of__inject__set,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys))
       => ( ( map_of(A,B,Xs) = map_of(A,B,Ys) )
        <=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Ys) ) ) ) ) ).

% map_of_inject_set
tff(fact_7407_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),A2: B,B2: A] :
      ( aa(set(option(A)),$o,finite_finite(option(A)),image(B,option(A),F3,top_top(set(B))))
     => aa(set(option(A)),$o,finite_finite(option(A)),image(B,option(A),fun_upd(B,option(A),F3,A2,aa(A,option(A),some(A),B2)),top_top(set(B)))) ) ).

% finite_range_updI
tff(fact_7408_inj__mapD,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F3),top_top(set(list(A))))
     => inj_on(A,B,F3,top_top(set(A))) ) ).

% inj_mapD
tff(fact_7409_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F3: fun(A,B),X: A] :
      aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F3),Xs))),X) = $ite(member(A,X,aa(list(A),set(A),set2(A),Xs)),aa(B,option(B),some(B),aa(A,B,F3,X)),none(B)) ).

% map_of_zip_map
tff(fact_7410_map__of__zip__upd,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B),Zs: list(A),Xb: B,Y: A,Z: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xs) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xs) )
       => ( ~ member(B,Xb,aa(list(B),set(B),set2(B),Xs))
         => ( ( fun_upd(B,option(A),map_of(B,A,zip(B,A,Xs,Ys)),Xb,aa(A,option(A),some(A),Y)) = fun_upd(B,option(A),map_of(B,A,zip(B,A,Xs,Zs)),Xb,aa(A,option(A),some(A),Z)) )
           => ( map_of(B,A,zip(B,A,Xs,Ys)) = map_of(B,A,zip(B,A,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_7411_map__fst__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),zip(A,B,Xs,Ys)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)),Xs) ).

% map_fst_zip_take
tff(fact_7412_map__snd__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(B),Ys: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),zip(B,A,Xs,Ys)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)),Ys) ).

% map_snd_zip_take
tff(fact_7413_map__of_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),Ps: list(product_prod(A,B))] : map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),P2),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P2),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P2))) ).

% map_of.simps(2)
tff(fact_7414_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: fun(A,B),Xaa: list(A),Y: A] :
          ( ( arg_min_list(A,B,Xb,Xaa) = Y )
         => ( ! [X4: A] :
                ( ( Xaa = aa(list(A),list(A),cons(A,X4),nil(A)) )
               => ( Y != X4 ) )
           => ( ! [X4: A,Y3: A,Zs2: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Zs2)) )
                 => ( Y != $let(
                        m2: A,
                        m2:= arg_min_list(A,B,Xb,aa(list(A),list(A),cons(A,Y3),Zs2)),
                        $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Xb,X4)),aa(A,B,Xb,m2)),X4,m2) ) ) )
             => ~ ( ( Xaa = nil(A) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_7415_inj__on__mapI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(list(A))] :
      ( inj_on(A,B,F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(list(A),set(A),set2(A),A5)))
     => inj_on(list(A),list(B),map(A,B,F3),A5) ) ).

% inj_on_mapI
tff(fact_7416_map__of__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_adf(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3))),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),map_of(A,C,Xs)) ).

% map_of_map
tff(fact_7417_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),Ta: list(product_prod(A,C)),K2: A,Xb: C] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,Ta),K2) = aa(C,option(C),some(C),Xb) )
       => ( aa(B,option(C),map_of(B,C,aa(list(product_prod(A,C)),list(product_prod(B,C)),map(product_prod(A,C),product_prod(B,C),aa(fun(A,fun(C,product_prod(B,C))),fun(product_prod(A,C),product_prod(B,C)),product_case_prod(A,C,product_prod(B,C)),aTP_Lamp_ado(fun(A,B),fun(A,fun(C,product_prod(B,C))),F3))),Ta)),aa(A,B,F3,K2)) = aa(C,option(C),some(C),Xb) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_7418_set__map__of__compr,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),$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,$o),aTP_Lamp_adp(list(product_prod(A,B)),fun(A,fun(B,$o)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_7419_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),Mb: fun(A,option(B)),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))
     => ( map_upds(A,B,Mb,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),nil(A))),Ys) = fun_upd(A,option(B),map_upds(A,B,Mb,Xs,Ys),Xb,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_7420_Id__on__fold,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( id_on(A,A5) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_adq(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A5) ) ) ).

% Id_on_fold
tff(fact_7421_Id__onI,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( member(A,A2,A5)
     => 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,A5)) ) ).

% Id_onI
tff(fact_7422_map__upds__apply__nontin,axiom,
    ! [B: $tType,A: $tType,Xb: A,Xs: list(A),F3: fun(A,option(B)),Ys: list(B)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,option(B),map_upds(A,B,F3,Xs,Ys),Xb) = aa(A,option(B),F3,Xb) ) ) ).

% map_upds_apply_nontin
tff(fact_7423_fun__upds__append__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Mb: fun(A,option(B)),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,Mb,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs),Ys) = map_upds(A,B,Mb,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_7424_fun__upds__append2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Mb: fun(A,option(B)),Zs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,Mb,Xs,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = map_upds(A,B,Mb,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_7425_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,Mb: fun(A,option(B)),Ys: list(B),Y: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( map_upds(A,B,Mb,Xs,list_update(B,Ys,I,Y)) = map_upds(A,B,Mb,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_7426_map__upds__Cons,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),A2: A,As3: list(A),B2: B,Bs: list(B)] : map_upds(A,B,Mb,aa(list(A),list(A),cons(A,A2),As3),aa(list(B),list(B),cons(B,B2),Bs)) = map_upds(A,B,fun_upd(A,option(B),Mb,A2,aa(B,option(B),some(B),B2)),As3,Bs) ).

% map_upds_Cons
tff(fact_7427_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A2: A,As3: list(A),Mb: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ member(A,A2,aa(list(A),set(A),set2(A),As3))
     => ( map_upds(A,B,fun_upd(A,option(B),Mb,A2,aa(B,option(B),some(B),B2)),As3,Bs) = fun_upd(A,option(B),map_upds(A,B,Mb,As3,Bs),A2,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_7428_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),Xss)) = concat(list(A),aa(list(A),list(list(list(A))),map(A,list(list(A)),aTP_Lamp_adr(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_7429_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,Nb),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_ads(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,Nb,Xs))) ).

% n_lists.simps(2)
tff(fact_7430_Id__on__set,axiom,
    ! [A: $tType,Xs: list(A)] : id_on(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_lx(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_7431_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_lx(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_7432_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_adt(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_7433_Id__on__iff,axiom,
    ! [A: $tType,Xb: A,Y: A,A5: set(A)] :
      ( 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,A5))
    <=> ( ( Xb = Y )
        & member(A,Xb,A5) ) ) ).

% Id_on_iff
tff(fact_7434_Id__on__eqI,axiom,
    ! [A: $tType,A2: A,B2: A,A5: set(A)] :
      ( ( A2 = B2 )
     => ( member(A,A2,A5)
       => 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,A5)) ) ) ).

% Id_on_eqI
tff(fact_7435_Id__onE,axiom,
    ! [A: $tType,C3: product_prod(A,A),A5: set(A)] :
      ( member(product_prod(A,A),C3,id_on(A,A5))
     => ~ ! [X4: A] :
            ( member(A,X4,A5)
           => ( C3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ) ) ).

% Id_onE
tff(fact_7436_distinct__set__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(set(A),aa(list(list(A)),list(set(A)),map(list(A),set(A),set2(A)),subseqs(A,Xs))) ) ).

% distinct_set_subseqs
tff(fact_7437_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xb: A,Ys: list(B)] : zip(A,B,replicate(A,Nb,Xb),Ys) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb)),take(B,Nb,Ys)) ).

% zip_replicate1
tff(fact_7438_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      subseqs(A,aa(list(A),list(A),cons(A,Xb),Xs)) = $let(
        xss: list(list(A)),
        xss:= subseqs(A,Xs),
        aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),aa(list(list(A)),list(list(A)),map(list(A),list(A),cons(A,Xb)),xss)),xss) ) ).

% subseqs.simps(2)
tff(fact_7439_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Nb: nat,Y: B] : zip(A,B,Xs,replicate(B,Nb,Y)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_mc(B,fun(A,product_prod(A,B))),Y)),take(A,Nb,Xs)) ).

% zip_replicate2
tff(fact_7440_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),cons(A,Xb),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb)),Ys)),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_7441_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      map_upds(A,B,fun_upd(A,option(B),F3,Xb,aa(B,option(B),some(B),Y)),Xs,Ys) = $ite(member(A,Xb,aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))),map_upds(A,B,F3,Xs,Ys),fun_upd(A,option(B),map_upds(A,B,F3,Xs,Ys),Xb,aa(B,option(B),some(B),Y))) ).

% map_upd_upds_conv_if
tff(fact_7442_Id__on__def,axiom,
    ! [A: $tType,A5: set(A)] : id_on(A,A5) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(A,set(product_prod(A,A)),aTP_Lamp_adu(A,set(product_prod(A,A))),A5)) ).

% Id_on_def
tff(fact_7443_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_adt(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_7444_transpose_Oelims,axiom,
    ! [A: $tType,Xb: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xb) = Y )
     => ( ( ( Xb = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( Xb = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2) )
             => ( Y != transpose(A,Xss2) ) )
         => ~ ! [X4: A,Xs2: list(A),Xss2: list(list(A))] :
                ( ( Xb = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X4),Xs2)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X4),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_adv(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_adw(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_7445_transpose__map__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(list(B))] : transpose(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F3)),Xs)) = aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F3)),transpose(B,Xs)) ).

% transpose_map_map
tff(fact_7446_transpose_Osimps_I2_J,axiom,
    ! [A: $tType,Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss)) = transpose(A,Xss) ).

% transpose.simps(2)
tff(fact_7447_transpose_Osimps_I1_J,axiom,
    ! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).

% transpose.simps(1)
tff(fact_7448_transpose__empty,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( transpose(A,Xs) = nil(list(A)) )
    <=> ! [X3: list(A)] :
          ( member(list(A),X3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( X3 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_7449_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_adv(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_adw(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_7450_transpose_Opelims,axiom,
    ! [A: $tType,Xb: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xb) = Y )
     => ( accp(list(list(A)),transpose_rel(A),Xb)
       => ( ( ( Xb = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ accp(list(list(A)),transpose_rel(A),nil(list(A))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( Xb = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2)) ) )
           => ~ ! [X4: A,Xs2: list(A),Xss2: list(list(A))] :
                  ( ( Xb = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X4),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X4),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_adv(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_adw(A,fun(list(A),list(list(A)))))),Xss2))))) )
                   => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X4),Xs2)),Xss2)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_7451_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Xss: list(list(A))] :
      ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),Xs)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_adv(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_adw(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_7452_transpose_Opsimps_I2_J,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss)) = transpose(A,Xss) ) ) ).

% transpose.psimps(2)
tff(fact_7453_transpose_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
     => ( transpose(A,nil(list(A))) = nil(list(A)) ) ) ).

% transpose.psimps(1)
tff(fact_7454_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),$o)] :
      ( accp(list(list(A)),transpose_rel(A),A0)
     => ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
         => aa(list(list(A)),$o,P,nil(list(A))) )
       => ( ! [Xss2: list(list(A))] :
              ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2))
             => ( aa(list(list(A)),$o,P,Xss2)
               => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2)) ) )
         => ( ! [X4: A,Xs2: list(A),Xss2: list(list(A))] :
                ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X4),Xs2)),Xss2))
               => ( aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_adw(A,fun(list(A),list(list(A)))))),Xss2))))
                 => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X4),Xs2)),Xss2)) ) )
           => aa(list(list(A)),$o,P,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_7455_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A),Y: A] :
          ( ( min_list(A,Xb) = Y )
         => ( accp(list(A),min_list_rel(A),Xb)
           => ( ! [X4: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),cons(A,X4),Xs2) )
                 => ( ( Y = aa(list(A),A,case_list(A,A,X4,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_abo(A,fun(list(A),fun(A,fun(list(A),A))),X4),Xs2)),Xs2) )
                   => ~ accp(list(A),min_list_rel(A),aa(list(A),list(A),cons(A,X4),Xs2)) ) )
             => ~ ( ( Xb = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_7456_remdups__adj_Opelims,axiom,
    ! [A: $tType,Xb: list(A),Y: list(A)] :
      ( ( remdups_adj(A,Xb) = Y )
     => ( accp(list(A),remdups_adj_rel(A),Xb)
       => ( ( ( Xb = nil(A) )
           => ( ( Y = nil(A) )
             => ~ accp(list(A),remdups_adj_rel(A),nil(A)) ) )
         => ( ! [X4: A] :
                ( ( Xb = aa(list(A),list(A),cons(A,X4),nil(A)) )
               => ( ( Y = aa(list(A),list(A),cons(A,X4),nil(A)) )
                 => ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),cons(A,X4),nil(A))) ) )
           => ~ ! [X4: A,Y3: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Xs2)) )
                 => ( ( Y = $ite(X4 = Y3,remdups_adj(A,aa(list(A),list(A),cons(A,X4),Xs2)),aa(list(A),list(A),cons(A,X4),remdups_adj(A,aa(list(A),list(A),cons(A,Y3),Xs2)))) )
                   => ~ accp(list(A),remdups_adj_rel(A),aa(list(A),list(A),cons(A,X4),aa(list(A),list(A),cons(A,Y3),Xs2))) ) ) ) ) ) ) ).

% remdups_adj.pelims
tff(fact_7457_list__encode_Opelims,axiom,
    ! [Xb: list(nat),Y: nat] :
      ( ( aa(list(nat),nat,nat_list_encode,Xb) = Y )
     => ( accp(list(nat),nat_list_encode_rel,Xb)
       => ( ( ( Xb = nil(nat) )
           => ( ( Y = zero_zero(nat) )
             => ~ accp(list(nat),nat_list_encode_rel,nil(nat)) ) )
         => ~ ! [X4: nat,Xs2: list(nat)] :
                ( ( Xb = aa(list(nat),list(nat),cons(nat,X4),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),X4),aa(list(nat),nat,nat_list_encode,Xs2)))) )
                 => ~ accp(list(nat),nat_list_encode_rel,aa(list(nat),list(nat),cons(nat,X4),Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_7458_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,Mb: fun(B,option(A)),Xb: B,Y: A,Z: A] :
      ( ( aa(B,option(A),Mb,Xb) = aa(A,option(A),some(A),Y) )
     => ( inj_on(B,option(A),Mb,dom(B,A,Mb))
       => ( ~ member(A,Z,ran(B,A,Mb))
         => ( ran(B,A,fun_upd(B,option(A),Mb,Xb,aa(A,option(A),some(A),Z))) = 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)),ran(B,A,Mb)),aa(set(A),set(A),insert(A,Y),bot_bot(set(A))))),aa(set(A),set(A),insert(A,Z),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_7459_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B))] :
      ( ( dom(A,B,F3) = bot_bot(set(A)) )
    <=> ! [X3: A] : aa(A,option(B),F3,X3) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_7460_fun__upd__None__if__notin__dom,axiom,
    ! [B: $tType,A: $tType,K2: A,Mb: fun(A,option(B))] :
      ( ~ member(A,K2,dom(A,B,Mb))
     => ( fun_upd(A,option(B),Mb,K2,none(B)) = Mb ) ) ).

% fun_upd_None_if_notin_dom
tff(fact_7461_dom__const,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : dom(A,B,aTP_Lamp_adx(fun(A,B),fun(A,option(B)),F3)) = top_top(set(A)) ).

% dom_const
tff(fact_7462_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_ph(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_7463_dom__map__of__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( dom(A,B,map_of(A,B,zip(A,B,Xs,Ys))) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% dom_map_of_zip
tff(fact_7464_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),Xb: A,Y: option(B)] :
      dom(A,B,fun_upd(A,option(B),F3,Xb,Y)) = $ite(Y = none(B),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F3)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))),aa(set(A),set(A),insert(A,Xb),dom(A,B,F3))) ).

% dom_fun_upd
tff(fact_7465_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),Xs: list(A),Ys: list(B)] : dom(A,B,map_upds(A,B,Mb,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),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))),dom(A,B,Mb)) ).

% dom_map_upds
tff(fact_7466_insert__dom,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),Xb: B,Y: A] :
      ( ( aa(B,option(A),F3,Xb) = aa(A,option(A),some(A),Y) )
     => ( aa(set(B),set(B),insert(B,Xb),dom(B,A,F3)) = dom(B,A,F3) ) ) ).

% insert_dom
tff(fact_7467_dom__minus,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),Xb: B,A5: set(B)] :
      ( ( aa(B,option(A),F3,Xb) = none(A) )
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F3)),aa(set(B),set(B),insert(B,Xb),A5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F3)),A5) ) ) ).

% dom_minus
tff(fact_7468_domI,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),Mb,A2) = aa(A,option(A),some(A),B2) )
     => member(B,A2,dom(B,A,Mb)) ) ).

% domI
tff(fact_7469_domD,axiom,
    ! [A: $tType,B: $tType,A2: A,Mb: fun(A,option(B))] :
      ( member(A,A2,dom(A,B,Mb))
     => ? [B4: B] : aa(A,option(B),Mb,A2) = aa(B,option(B),some(B),B4) ) ).

% domD
tff(fact_7470_domIff,axiom,
    ! [A: $tType,B: $tType,A2: A,Mb: fun(A,option(B))] :
      ( member(A,A2,dom(A,B,Mb))
    <=> ( aa(A,option(B),Mb,A2) != none(B) ) ) ).

% domIff
tff(fact_7471_dom__def,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B))] : dom(A,B,Mb) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ady(fun(A,option(B)),fun(A,$o),Mb)) ).

% dom_def
tff(fact_7472_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B))] :
      ( aa(set(A),$o,finite_finite(A),dom(A,B,F3))
     => ( ~ aa(set(A),$o,finite_finite(A),top_top(set(A)))
       => ? [X4: A] : aa(A,option(B),F3,X4) = none(B) ) ) ).

% finite_map_freshness
tff(fact_7473_dom__map__of__conv__image__fst,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] : dom(A,B,map_of(A,B,Xys)) = image(product_prod(A,B),A,product_fst(A,B),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ).

% dom_map_of_conv_image_fst
tff(fact_7474_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),P: fun(fun(A,option(B)),$o)] :
      ( aa(set(A),$o,finite_finite(A),dom(A,B,Mb))
     => ( aa(fun(A,option(B)),$o,P,aTP_Lamp_ph(A,option(B)))
       => ( ! [K: A,V3: B,M: fun(A,option(B))] :
              ( aa(set(A),$o,finite_finite(A),dom(A,B,M))
             => ( ~ member(A,K,dom(A,B,M))
               => ( aa(fun(A,option(B)),$o,P,M)
                 => aa(fun(A,option(B)),$o,P,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V3))) ) ) )
         => aa(fun(A,option(B)),$o,P,Mb) ) ) ) ).

% finite_Map_induct
tff(fact_7475_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),Xb: A] :
      ( ( dom(A,B,F3) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) )
    <=> ? [V5: B] : F3 = fun_upd(A,option(B),aTP_Lamp_ph(A,option(B)),Xb,aa(B,option(B),some(B),V5)) ) ).

% dom_eq_singleton_conv
tff(fact_7476_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Mb: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,Mb) )
     => ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_adz(fun(A,option(B)),fun(A,product_prod(A,B)),Mb)),Xs)) = Mb ) ) ).

% map_of_map_keys
tff(fact_7477_dom__override__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),G3: fun(A,option(B)),A5: set(A)] : dom(A,B,override_on(A,option(B),F3,G3,A5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F3)),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aea(fun(A,option(B)),fun(set(A),fun(A,$o)),G3),A5)))),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aeb(fun(A,option(B)),fun(set(A),fun(A,$o)),G3),A5))) ).

% dom_override_on
tff(fact_7478_set__relcomp,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_aed(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_7479_relpow_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,R: 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)),R) = 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),R),R) ).

% relpow.simps(2)
tff(fact_7480_relpow__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,R: 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)),R) = 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),R),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),R)) ).

% relpow_add
tff(fact_7481_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A2: A,C3: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),C3),relcomp(A,C,B,R2,S2))
     => ~ ! [B4: C] :
            ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A2),B4),R2)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),C3),S2) ) ) ).

% relcompEpair
tff(fact_7482_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( member(product_prod(A,B),Xz,relcomp(A,C,B,R2,S2))
     => ~ ! [X4: A,Y3: C,Z3: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z3) )
           => ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X4),Y3),R2)
             => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Z3),S2) ) ) ) ).

% relcompE
tff(fact_7483_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R2: set(product_prod(A,B)),C3: C,S2: set(product_prod(B,C))] :
      ( 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),R2)
     => ( member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3),S2)
       => member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A2),C3),relcomp(A,B,C,R2,S2)) ) ) ).

% relcomp.relcompI
tff(fact_7484_relcomp_Osimps,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( 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,R2,S2))
    <=> ? [A6: A,B6: C,C4: B] :
          ( ( A1 = A6 )
          & ( A22 = C4 )
          & member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A6),B6),R2)
          & member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B6),C4),S2) ) ) ).

% relcomp.simps
tff(fact_7485_relcomp_Ocases,axiom,
    ! [A: $tType,B: $tType,C: $tType,A1: A,A22: B,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] :
      ( 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,R2,S2))
     => ~ ! [B4: C] :
            ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),B4),R2)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),A22),S2) ) ) ).

% relcomp.cases
tff(fact_7486_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S2: set(product_prod(C,B))] : relcomp(A,C,B,R2,S2) = aa(fun(product_prod(A,B),$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_aee(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),R2),S2))) ).

% relcomp_unfold
tff(fact_7487_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),R)
     => ( aa(set(product_prod(B,C)),$o,finite_finite(product_prod(B,C)),S3)
       => ( relcomp(A,B,C,R,S3) = 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_aeg(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S3)),bot_bot(set(product_prod(A,C))),R) ) ) ) ).

% relcomp_fold
tff(fact_7488_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B)),Xb: product_prod(C,A),R: set(product_prod(C,A))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),S3)
     => ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert(product_prod(C,A),Xb),R),S3) = 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_aeh(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xb)),relcomp(C,A,B,R,S3),S3) ) ) ).

% insert_relcomp_fold
tff(fact_7489_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B)),Xb: product_prod(C,A),X6: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),S3)
     => ( 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)))),S3)),X6) = 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_aeh(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xb)),X6,S3) ) ) ).

% insert_relcomp_union_fold
tff(fact_7490_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),K2: A,V: B] : graph(A,B,fun_upd(A,option(B),Mb,K2,aa(B,option(B),some(B),V))) = 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),K2),V)),graph(A,B,fun_upd(A,option(B),Mb,K2,none(B)))) ).

% graph_map_upd
tff(fact_7491_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_aco(fun(A,B),fun(A,product_prod(A,B)),F3)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F3),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_7492_restrict__out,axiom,
    ! [A: $tType,B: $tType,Xb: A,A5: set(A),Mb: fun(A,option(B))] :
      ( ~ member(A,Xb,A5)
     => ( aa(A,option(B),restrict_map(A,B,Mb,A5),Xb) = none(B) ) ) ).

% restrict_out
tff(fact_7493_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D4: set(A),X: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_ph(A,option(B)),D4),X) = none(B) ).

% restrict_map_empty
tff(fact_7494_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),X: A] : aa(A,option(B),restrict_map(A,B,Mb,bot_bot(set(A))),X) = none(B) ).

% restrict_map_to_empty
tff(fact_7495_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_ph(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_7496_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),Xb: A,Y: option(B),D4: set(A)] :
      restrict_map(A,B,fun_upd(A,option(B),Mb,Xb,Y),D4) = $ite(member(A,Xb,D4),fun_upd(A,option(B),restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),Xb,Y),restrict_map(A,B,Mb,D4)) ).

% restrict_fun_upd
tff(fact_7497_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,Xb: A,D4: set(A),Mb: fun(A,option(B)),Y: option(B)] :
      ( member(A,Xb,D4)
     => ( fun_upd(A,option(B),restrict_map(A,B,Mb,D4),Xb,Y) = fun_upd(A,option(B),restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),Xb,Y) ) ) ).

% fun_upd_restrict_conv
tff(fact_7498_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),D4: set(A),Xb: A] :
      fun_upd(A,option(B),restrict_map(A,B,Mb,D4),Xb,none(B)) = $ite(member(A,Xb,D4),restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),restrict_map(A,B,Mb,D4)) ).

% fun_upd_None_restrict
tff(fact_7499_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),D4: set(A),Mb: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D4)
       => ( restrict_map(A,B,map_upds(A,B,Mb,Xs,Ys),D4) = map_upds(A,B,restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(list(A),set(A),set2(A),Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_7500_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),Xb: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),Mb,Xb,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))) = restrict_map(A,B,Mb,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_7501_ran__restrictD,axiom,
    ! [B: $tType,A: $tType,Y: A,Mb: fun(B,option(A)),A5: set(B)] :
      ( member(A,Y,ran(B,A,restrict_map(B,A,Mb,A5)))
     => ? [X4: B] :
          ( member(B,X4,A5)
          & ( aa(B,option(A),Mb,X4) = aa(A,option(A),some(A),Y) ) ) ) ).

% ran_restrictD
tff(fact_7502_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K2: A,V: B,Mb: fun(A,option(B)),A5: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V),graph(A,B,restrict_map(A,B,Mb,A5)))
     => member(A,K2,A5) ) ).

% graph_restrictD(1)
tff(fact_7503_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K2: A,V: B,Mb: fun(A,option(B)),A5: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V),graph(A,B,restrict_map(A,B,Mb,A5)))
     => ( aa(A,option(B),Mb,K2) = aa(B,option(B),some(B),V) ) ) ).

% graph_restrictD(2)
tff(fact_7504_in__graphI,axiom,
    ! [A: $tType,B: $tType,Mb: fun(B,option(A)),K2: B,V: A] :
      ( ( aa(B,option(A),Mb,K2) = aa(A,option(A),some(A),V) )
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K2),V),graph(B,A,Mb)) ) ).

% in_graphI
tff(fact_7505_in__graphD,axiom,
    ! [A: $tType,B: $tType,K2: A,V: B,Mb: fun(A,option(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K2),V),graph(A,B,Mb))
     => ( aa(A,option(B),Mb,K2) = aa(B,option(B),some(B),V) ) ) ).

% in_graphD
tff(fact_7506_restrict__map__def,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),A5: set(A),X: A] :
      aa(A,option(B),restrict_map(A,B,Mb,A5),X) = $ite(member(A,X,A5),aa(A,option(B),Mb,X),none(B)) ).

% restrict_map_def
tff(fact_7507_graph__def,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B))] : graph(A,B,Mb) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_aei(fun(A,option(B)),fun(product_prod(A,B),$o),Mb)) ).

% graph_def
tff(fact_7508_graph__fun__upd__None,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B)),K2: A] : graph(A,B,fun_upd(A,option(B),Mb,K2,none(B))) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(A,fun(product_prod(A,B),$o),aTP_Lamp_aej(fun(A,option(B)),fun(A,fun(product_prod(A,B),$o)),Mb),K2)) ).

% graph_fun_upd_None
tff(fact_7509_graph__eq__to__snd__dom,axiom,
    ! [B: $tType,A: $tType,Mb: fun(A,option(B))] : graph(A,B,Mb) = image(A,product_prod(A,B),aTP_Lamp_adz(fun(A,option(B)),fun(A,product_prod(A,B)),Mb),dom(A,B,Mb)) ).

% graph_eq_to_snd_dom
tff(fact_7510_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),D4: set(A),Xb: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,Mb,D4),Xb,Y) = fun_upd(A,option(B),restrict_map(A,B,Mb,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),Xb,Y) ).

% fun_upd_restrict
tff(fact_7511_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),Xb: A] : restrict_map(A,B,F3,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))) = fun_upd(A,option(B),F3,Xb,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_7512_graph__map__of__if__distinct__dom,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
     => ( graph(A,B,map_of(A,B,Al)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al) ) ) ).

% graph_map_of_if_distinct_dom
tff(fact_7513_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite(product_prod(A,B)),S3)
     => 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_ael(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S3))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_7514_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(list(nat),list($o),map(nat,$o,bit_se5641148757651400278ts_bit(A,A2)),upt(zero_zero(nat),Nb))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_7515_remdups__upt,axiom,
    ! [Mb: nat,Nb: nat] : remdups(nat,upt(Mb,Nb)) = upt(Mb,Nb) ).

% remdups_upt
tff(fact_7516_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(list(nat),nat,hd(nat),upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_7517_drop__upt,axiom,
    ! [Mb: nat,I: nat,J: nat] : drop(nat,Mb,upt(I,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Mb),J) ).

% drop_upt
tff(fact_7518_length__upt,axiom,
    ! [I: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) ).

% length_upt
tff(fact_7519_take__upt,axiom,
    ! [I: 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),I),Mb)),Nb)
     => ( take(nat,Mb,upt(I,Nb)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Mb)) ) ) ).

% take_upt
tff(fact_7520_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_7521_sorted__list__of__set__range,axiom,
    ! [Mb: nat,Nb: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or7035219750837199246ssThan(nat,Mb,Nb)) = upt(Mb,Nb) ).

% sorted_list_of_set_range
tff(fact_7522_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_7523_nth__upt,axiom,
    ! [I: nat,K2: 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),I),K2)),J)
     => ( aa(nat,nat,nth(nat,upt(I,J)),K2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2) ) ) ).

% nth_upt
tff(fact_7524_map__fst__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Nb,Xs)) = upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_7525_upt__rec__numeral,axiom,
    ! [Mb: num,Nb: num] :
      upt(aa(num,nat,numeral_numeral(nat),Mb),aa(num,nat,numeral_numeral(nat),Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),Mb)),aa(num,nat,numeral_numeral(nat),Nb)),aa(list(nat),list(nat),cons(nat,aa(num,nat,numeral_numeral(nat),Mb)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Mb)),aa(num,nat,numeral_numeral(nat),Nb))),nil(nat)) ).

% upt_rec_numeral
tff(fact_7526_atMost__upto,axiom,
    ! [Nb: nat] : set_ord_atMost(nat,Nb) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,Nb))) ).

% atMost_upto
tff(fact_7527_atLeast__upt,axiom,
    ! [Nb: nat] : set_ord_lessThan(nat,Nb) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),Nb)) ).

% atLeast_upt
tff(fact_7528_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_7529_upt__conv__Cons__Cons,axiom,
    ! [Mb: nat,Nb: nat,Ns: list(nat),Q3: nat] :
      ( ( aa(list(nat),list(nat),cons(nat,Mb),aa(list(nat),list(nat),cons(nat,Nb),Ns)) = upt(Mb,Q3) )
    <=> ( aa(list(nat),list(nat),cons(nat,Nb),Ns) = upt(aa(nat,nat,suc,Mb),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_7530_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Mb: nat] : set_or5935395276787703475ssThan(nat,Nb,Mb) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Nb),Mb)) ).

% greaterThanLessThan_upt
tff(fact_7531_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : set_or3652927894154168847AtMost(nat,Nb,Mb) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Nb),aa(nat,nat,suc,Mb))) ).

% greaterThanAtMost_upt
tff(fact_7532_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or7035219750837199246ssThan(nat,I,J) = aa(list(nat),set(nat),set2(nat),upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_7533_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : set_or1337092689740270186AtMost(nat,Nb,Mb) = aa(list(nat),set(nat),set2(nat),upt(Nb,aa(nat,nat,suc,Mb))) ).

% atLeastAtMost_upt
tff(fact_7534_distinct__upt,axiom,
    ! [I: nat,J: nat] : distinct(nat,upt(I,J)) ).

% distinct_upt
tff(fact_7535_map__replicate__trivial,axiom,
    ! [A: $tType,Xb: A,I: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_aem(A,fun(nat,A),Xb)),upt(zero_zero(nat),I)) = replicate(A,I,Xb) ).

% map_replicate_trivial
tff(fact_7536_map__Suc__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa(list(nat),list(nat),map(nat,nat,suc),upt(Mb,Nb)) = upt(aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb)) ).

% map_Suc_upt
tff(fact_7537_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_aen(nat,fun(nat,nat),Nb)),upt(zero_zero(nat),Mb)) = upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) ).

% map_add_upt
tff(fact_7538_enumerate__map__upt,axiom,
    ! [A: $tType,Nb: nat,F3: fun(nat,A),Mb: nat] : enumerate(A,Nb,aa(list(nat),list(A),map(nat,A,F3),upt(Nb,Mb))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_aeo(fun(nat,A),fun(nat,product_prod(nat,A)),F3)),upt(Nb,Mb)) ).

% enumerate_map_upt
tff(fact_7539_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( upt(I,J) = aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_7540_enumerate__eq__zip,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : enumerate(A,Nb,Xs) = zip(nat,A,upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_7541_map__upt__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),Nb: nat] : aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(list(A),list(A),cons(A,aa(nat,A,F3,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_xw(fun(nat,A),fun(nat,A),F3)),upt(zero_zero(nat),Nb))) ).

% map_upt_Suc
tff(fact_7542_map__decr__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_ma(nat,nat)),upt(aa(nat,nat,suc,Mb),aa(nat,nat,suc,Nb))) = upt(Mb,Nb) ).

% map_decr_upt
tff(fact_7543_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(nat),list(A),map(nat,A,nth(A,Xs)),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_7544_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K2))) ) ) ).

% upt_add_eq_append
tff(fact_7545_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Nb: nat,Mb: nat,F3: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F3),upt(Mb,Nb))),I) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I)) ) ) ).

% nth_map_upt
tff(fact_7546_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),cons(nat,Xb),Xs) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = Xb )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_7547_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J),aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)),nil(nat)) ).

% upt_rec
tff(fact_7548_enumerate__replicate__eq,axiom,
    ! [A: $tType,Nb: nat,Mb: nat,A2: A] : enumerate(A,Nb,replicate(A,Mb,A2)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_aep(A,fun(nat,product_prod(nat,A)),A2)),upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Mb))) ).

% enumerate_replicate_eq
tff(fact_7549_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Mb: nat,F3: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),Mb) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),I2)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F3),upt(Mb,Nb)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_7550_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),cons(nat,J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_7551_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J),aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),cons(nat,J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_7552_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),B5)
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_aeq(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B5)) ) ).

% comp_fun_commute_product_fold
tff(fact_7553_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( Nb = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = Nb ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_aes(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),Nb)) ) ) ) ).

% transpose_rectangle
tff(fact_7554_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_adv(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aet(list(A),$o)),Xss)) ).

% transpose_aux_filter_head
tff(fact_7555_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(nat,nat,foldr(list(A),nat,aTP_Lamp_aeu(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_7556_filter__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),filter2(A,Q),Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aev(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),Xs) ).

% filter_filter
tff(fact_7557_filter__True,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) )
     => ( aa(list(A),list(A),filter2(A,P),Xs) = Xs ) ) ).

% filter_True
tff(fact_7558_filter__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,P),Xs)),aa(list(A),list(A),filter2(A,P),Ys)) ).

% filter_append
tff(fact_7559_foldr__append,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),Xs: list(B),Ys: list(B),A2: A] : aa(A,A,foldr(B,A,F3,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),A2) = aa(A,A,foldr(B,A,F3,Xs),aa(A,A,foldr(B,A,F3,Ys),A2)) ).

% foldr_append
tff(fact_7560_remove1__filter__not,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      ( ~ aa(A,$o,P,Xb)
     => ( remove1(A,Xb,aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ).

% remove1_filter_not
tff(fact_7561_removeAll__filter__not,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      ( ~ aa(A,$o,P,Xb)
     => ( aa(list(A),list(A),removeAll(A,Xb),aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ).

% removeAll_filter_not
tff(fact_7562_set__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_aew(fun(A,$o),fun(list(A),fun(A,$o)),P),Xs)) ).

% set_filter
tff(fact_7563_filter__False,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X4) )
     => ( aa(list(A),list(A),filter2(A,P),Xs) = nil(A) ) ) ).

% filter_False
tff(fact_7564_foldr__replicate,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Nb: nat,Xb: B] : foldr(B,A,F3,replicate(B,Nb,Xb)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Nb),aa(B,fun(A,A),F3,Xb)) ).

% foldr_replicate
tff(fact_7565_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),aa(list(B),list(A),map(B,A,F3),Xs))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P),F3)),Xs)) ).

% length_filter_map
tff(fact_7566_filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),filter2(A,P),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P),F3)),Xs)) ).

% filter_map
tff(fact_7567_foldr__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: fun(B,fun(A,A)),F3: fun(C,B),Xs: list(C),A2: A] : aa(A,A,foldr(B,A,G3,aa(list(C),list(B),map(C,B,F3),Xs)),A2) = aa(A,A,foldr(C,A,aa(fun(C,B),fun(C,fun(A,A)),comp(B,fun(A,A),C,G3),F3),Xs),A2) ).

% foldr_map
tff(fact_7568_filter__shuffles,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] : image(list(A),list(A),filter2(A,P),shuffles(A,Xs,Ys)) = shuffles(A,aa(list(A),list(A),filter2(A,P),Xs),aa(list(A),list(A),filter2(A,P),Ys)) ).

% filter_shuffles
tff(fact_7569_filter__concat,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(list(A))] : aa(list(A),list(A),filter2(A,P2),concat(A,Xs)) = concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),filter2(A,P2)),Xs)) ).

% filter_concat
tff(fact_7570_distinct__map__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),P: fun(B,$o)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
     => distinct(A,aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs))) ) ).

% distinct_map_filter
tff(fact_7571_distinct__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),filter2(A,P),Xs)) ) ).

% distinct_filter
tff(fact_7572_filter__eq__Cons__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Ys: list(A),Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Ys) = aa(list(A),list(A),cons(A,Xb),Xs) )
    <=> ? [Us2: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),cons(A,Xb),Vs3)) )
          & ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X3) )
          & aa(A,$o,P,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs3) ) ) ) ).

% filter_eq_Cons_iff
tff(fact_7573_Cons__eq__filter__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),cons(A,Xb),Xs) = aa(list(A),list(A),filter2(A,P),Ys) )
    <=> ? [Us2: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),cons(A,Xb),Vs3)) )
          & ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X3) )
          & aa(A,$o,P,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs3) ) ) ) ).

% Cons_eq_filter_iff
tff(fact_7574_filter__eq__ConsD,axiom,
    ! [A: $tType,P: fun(A,$o),Ys: list(A),Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Ys) = aa(list(A),list(A),cons(A,Xb),Xs) )
     => ? [Us3: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),cons(A,Xb),Vs2)) )
          & ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P,X) )
          & aa(A,$o,P,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs2) ) ) ) ).

% filter_eq_ConsD
tff(fact_7575_Cons__eq__filterD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),cons(A,Xb),Xs) = aa(list(A),list(A),filter2(A,P),Ys) )
     => ? [Us3: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),cons(A,Xb),Vs2)) )
          & ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P,X) )
          & aa(A,$o,P,Xb)
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs2) ) ) ) ).

% Cons_eq_filterD
tff(fact_7576_filter__empty__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xs) = nil(A) )
    <=> ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X3) ) ) ).

% filter_empty_conv
tff(fact_7577_empty__filter__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),filter2(A,P),Xs) )
    <=> ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X3) ) ) ).

% empty_filter_conv
tff(fact_7578_filter__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys )
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,P,X4)
            <=> aa(A,$o,Q,X4) ) )
       => ( aa(list(A),list(A),filter2(A,P),Xs) = aa(list(A),list(A),filter2(A,Q),Ys) ) ) ) ).

% filter_cong
tff(fact_7579_filter__id__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xs) = Xs )
    <=> ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) ) ) ).

% filter_id_conv
tff(fact_7580_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list(B),K2: list(B),F3: fun(B,fun(A,A)),G3: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( L = K2 )
       => ( ! [A4: A,X4: B] :
              ( member(B,X4,aa(list(B),set(B),set2(B),L))
             => ( aa(A,A,aa(B,fun(A,A),F3,X4),A4) = aa(A,A,aa(B,fun(A,A),G3,X4),A4) ) )
         => ( aa(A,A,foldr(B,A,F3,L),A2) = aa(A,A,foldr(B,A,G3,K2),B2) ) ) ) ) ).

% foldr_cong
tff(fact_7581_replicate__length__filter,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Xb)),Xs)),Xb) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Xb)),Xs) ).

% replicate_length_filter
tff(fact_7582_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aTP_Lamp_aex(fun(A,$o),fun(A,$o),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_7583_filter_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),cons(A,Xb),Xs)) = $ite(aa(A,$o,P,Xb),aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),filter2(A,P),Xs)),aa(list(A),list(A),filter2(A,P),Xs)) ).

% filter.simps(2)
tff(fact_7584_filter__insort__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),Xb: A,F3: fun(A,B),Xs: list(A)] :
          ( ~ aa(A,$o,P,Xb)
         => ( aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),Xb),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ) ).

% filter_insort_triv
tff(fact_7585_filter__remove1,axiom,
    ! [A: $tType,Q: fun(A,$o),Xb: A,Xs: list(A)] : aa(list(A),list(A),filter2(A,Q),remove1(A,Xb,Xs)) = remove1(A,Xb,aa(list(A),list(A),filter2(A,Q),Xs)) ).

% filter_remove1
tff(fact_7586_removeAll__filter__not__eq,axiom,
    ! [A: $tType,Xb: A] : removeAll(A,Xb) = filter2(A,aTP_Lamp_aey(A,fun(A,$o),Xb)) ).

% removeAll_filter_not_eq
tff(fact_7587_partition__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] : member(list(A),Xs,shuffles(A,aa(list(A),list(A),filter2(A,P),Xs),aa(list(A),list(A),filter2(A,aTP_Lamp_aex(fun(A,$o),fun(A,$o),P)),Xs))) ).

% partition_in_shuffles
tff(fact_7588_remdups__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : remdups(A,aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),remdups(A,Xs)) ).

% remdups_filter
tff(fact_7589_filter__replicate,axiom,
    ! [A: $tType,P: fun(A,$o),Nb: nat,Xb: A] :
      aa(list(A),list(A),filter2(A,P),replicate(A,Nb,Xb)) = $ite(aa(A,$o,P,Xb),replicate(A,Nb,Xb),nil(A)) ).

% filter_replicate
tff(fact_7590_filter_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(list(A),list(A),filter2(A,P),nil(A)) = nil(A) ).

% filter.simps(1)
tff(fact_7591_length__filter__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_filter_less
tff(fact_7592_inter__set__filter,axiom,
    ! [A: $tType,A5: set(A),Xs: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_a(set(A),fun(A,$o),A5)),Xs)) ).

% inter_set_filter
tff(fact_7593_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,$o),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),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% filter_is_subset
tff(fact_7594_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_filter_le
tff(fact_7595_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),insert(A,Y),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_aez(fun(A,B),fun(A,fun(A,$o)),F3),Y)),Xs) = aa(list(A),list(A),filter2(A,aa(A,fun(A,$o),fequal(A),Y)),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_7596_foldr__Nil,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A))] : foldr(B,A,F3,nil(B)) = id(A) ).

% foldr_Nil
tff(fact_7597_foldr__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),P: fun(B,$o),Xs: list(B)] : foldr(B,A,F3,aa(list(B),list(B),filter2(B,P),Xs)) = foldr(B,A,aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_afa(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),F3),P),Xs) ).

% foldr_filter
tff(fact_7598_foldr__Cons,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xb: B,Xs: list(B)] : foldr(B,A,F3,aa(list(B),list(B),cons(B,Xb),Xs)) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(B,fun(A,A),F3,Xb)),foldr(B,A,F3,Xs)) ).

% foldr_Cons
tff(fact_7599_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_afb(list(B),fun(nat,nat)),Xss),zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,foldr(list(B),nat,aTP_Lamp_afc(list(B),fun(nat,nat)),aa(list(list(B)),list(list(B)),filter2(list(B),aTP_Lamp_afd(list(B),$o)),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_7600_filter__in__nths,axiom,
    ! [A: $tType,Xs: list(A),S2: set(nat)] :
      ( distinct(A,Xs)
     => ( aa(list(A),list(A),filter2(A,aa(set(nat),fun(A,$o),aTP_Lamp_afe(list(A),fun(set(nat),fun(A,$o)),Xs),S2)),Xs) = nths(A,Xs,S2) ) ) ).

% filter_in_nths
tff(fact_7601_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_aeu(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aet(list(A),$o)),Xs)) ).

% transpose_max_length
tff(fact_7602_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_aff(A,fun(A,$o),Y)),Xs)) ).

% set_minus_filter_out
tff(fact_7603_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_afg(list(A),fun(A,$o),Xs)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_7604_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_afh(list(A),fun(A,$o),Xs)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_7605_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_afg(list(A),fun(A,$o),Ys)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_7606_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_afh(list(A),fun(A,$o),Ys)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_7607_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_afi(fun(A,$o),fun(list(A),fun(nat,$o)),P),Xs))) ).

% filter_eq_nths
tff(fact_7608_length__filter__conv__card,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P2),Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_afi(fun(A,$o),fun(list(A),fun(nat,$o)),P2),Xs))) ).

% length_filter_conv_card
tff(fact_7609_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs)) = aa(set(A),nat,finite_card(A),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(list(A),set(A),set2(A),Xs))) ) ) ).

% distinct_length_filter
tff(fact_7610_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_afj(fun(B,A),fun(A,fun(B,fun(A,A))),F3),A2),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_7611_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_afk(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afl(nat,fun(list(A),$o),I)),Xs)) ) ) ).

% nth_transpose
tff(fact_7612_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),Xs: list(B)] : map_filter(B,A,F3,Xs) = aa(list(B),list(A),map(B,A,aa(fun(B,option(A)),fun(B,A),comp(option(A),A,B,the2(A)),F3)),aa(list(B),list(B),filter2(B,aTP_Lamp_afm(fun(B,option(A)),fun(B,$o),F3)),Xs)) ).

% map_filter_def
tff(fact_7613_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_adw(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aet(list(A),$o)),Xss)) ).

% transpose_aux_filter_tail
tff(fact_7614_tl__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa(list(nat),list(nat),tl(nat),upt(Mb,Nb)) = upt(aa(nat,nat,suc,Mb),Nb) ).

% tl_upt
tff(fact_7615_tl__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xs)),Ys) ) ) ).

% tl_append2
tff(fact_7616_remdups__adj__Cons__alt,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),tl(A),remdups_adj(A,aa(list(A),list(A),cons(A,Xb),Xs)))) = remdups_adj(A,aa(list(A),list(A),cons(A,Xb),Xs)) ).

% remdups_adj_Cons_alt
tff(fact_7617_length__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(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)) ).

% length_tl
tff(fact_7618_list_Ocollapse,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) = List ) ) ).

% list.collapse
tff(fact_7619_hd__Cons__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),Xs)),aa(list(A),list(A),tl(A),Xs)) = Xs ) ) ).

% hd_Cons_tl
tff(fact_7620_tl__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),list(A),tl(A),replicate(A,Nb,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Xb) ).

% tl_replicate
tff(fact_7621_list_Omap__sel_I2_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F3: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),list(B),tl(B),aa(list(A),list(B),map(A,B,F3),A2)) = aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),tl(A),A2)) ) ) ).

% list.map_sel(2)
tff(fact_7622_map__tl,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),tl(B),Xs)) = aa(list(A),list(A),tl(A),aa(list(B),list(A),map(B,A,F3),Xs)) ).

% map_tl
tff(fact_7623_distinct__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),tl(A),Xs)) ) ).

% distinct_tl
tff(fact_7624_list_Oset__sel_I2_J,axiom,
    ! [A: $tType,A2: list(A),Xb: A] :
      ( ( A2 != nil(A) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),A2)))
       => member(A,Xb,aa(list(A),set(A),set2(A),A2)) ) ) ).

% list.set_sel(2)
tff(fact_7625_take__tl,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : take(A,Nb,aa(list(A),list(A),tl(A),Xs)) = aa(list(A),list(A),tl(A),take(A,aa(nat,nat,suc,Nb),Xs)) ).

% take_tl
tff(fact_7626_drop__Suc,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,aa(nat,nat,suc,Nb),Xs) = drop(A,Nb,aa(list(A),list(A),tl(A),Xs)) ).

% drop_Suc
tff(fact_7627_tl__drop,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),drop(A,Nb,Xs)) = drop(A,Nb,aa(list(A),list(A),tl(A),Xs)) ).

% tl_drop
tff(fact_7628_list_Oexpand,axiom,
    ! [A: $tType,List: list(A),List2: list(A)] :
      ( ( ( List = nil(A) )
      <=> ( List2 = nil(A) ) )
     => ( ( ( List != nil(A) )
         => ( ( List2 != nil(A) )
           => ( ( aa(list(A),A,hd(A),List) = aa(list(A),A,hd(A),List2) )
              & ( aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),tl(A),List2) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
tff(fact_7629_map__of__filter__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K2: B,Z: A,P: fun(B,fun(A,$o))] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K2) = aa(A,option(A),some(A),Z) )
     => ( aa(A,$o,aa(B,fun(A,$o),P,K2),Z)
       => ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),filter2(product_prod(B,A),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P)),Xs)),K2) = aa(A,option(A),some(A),Z) ) ) ) ).

% map_of_filter_in
tff(fact_7630_Nil__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),tl(A),Xs) )
    <=> ( ( Xs = nil(A) )
        | ? [X3: A] : Xs = aa(list(A),list(A),cons(A,X3),nil(A)) ) ) ).

% Nil_tl
tff(fact_7631_tl__Nil,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ? [X3: A] : Xs = aa(list(A),list(A),cons(A,X3),nil(A)) ) ) ).

% tl_Nil
tff(fact_7632_list_Osel_I2_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).

% list.sel(2)
tff(fact_7633_list_Osel_I3_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),cons(A,X21),X222)) = X222 ).

% list.sel(3)
tff(fact_7634_map__filter__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,option(A))] : map_filter(B,A,F3,nil(B)) = nil(A) ).

% map_filter_simps(2)
tff(fact_7635_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),tl(A),Ys),aTP_Lamp_afn(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_7636_tl__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_afo(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_7637_list_Oexhaust__sel,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( List = aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) ) ) ).

% list.exhaust_sel
tff(fact_7638_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),Xb: B,Xs: list(B)] : map_filter(B,A,F3,aa(list(B),list(B),cons(B,Xb),Xs)) = case_option(list(A),A,map_filter(B,A,F3,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_afp(fun(B,option(A)),fun(list(B),fun(A,list(A))),F3),Xs),aa(B,option(A),F3,Xb)) ).

% map_filter_simps(1)
tff(fact_7639_tl__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),take(A,Nb,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),aa(list(A),list(A),tl(A),Xs)) ).

% tl_take
tff(fact_7640_list_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      aa(list(B),A,case_list(A,B,F1,F22),List) = $ite(List = nil(B),F1,aa(list(B),A,aa(B,fun(list(B),A),F22,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ).

% list.case_eq_if
tff(fact_7641_concat__conv__foldr,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,Xss) = aa(list(A),list(A),foldr(list(A),list(A),append(A),Xss),nil(A)) ).

% concat_conv_foldr
tff(fact_7642_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      aa(list(A),nat,size_size(list(A)),Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)))) ).

% Nitpick.size_list_simp(2)
tff(fact_7643_nth__tl,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)),aa(list(A),list(A),tl(A),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,Nb)) ) ) ).

% nth_tl
tff(fact_7644_remdups__adj__append,axiom,
    ! [A: $tType,Xs_1: list(A),Xb: A,Xs_2: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),cons(A,Xb),Xs_2))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),cons(A,Xb),nil(A))))),aa(list(A),list(A),tl(A),remdups_adj(A,aa(list(A),list(A),cons(A,Xb),Xs_2)))) ).

% remdups_adj_append
tff(fact_7645_Cons__in__shuffles__iff,axiom,
    ! [A: $tType,Z: A,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( member(list(A),aa(list(A),list(A),cons(A,Z),Zs),shuffles(A,Xs,Ys))
    <=> ( ( ( Xs != nil(A) )
          & ( aa(list(A),A,hd(A),Xs) = Z )
          & member(list(A),Zs,shuffles(A,aa(list(A),list(A),tl(A),Xs),Ys)) )
        | ( ( Ys != nil(A) )
          & ( aa(list(A),A,hd(A),Ys) = Z )
          & member(list(A),Zs,shuffles(A,Xs,aa(list(A),list(A),tl(A),Ys))) ) ) ) ).

% Cons_in_shuffles_iff
tff(fact_7646_list_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,P,aa(list(B),A,case_list(A,B,F1,F22),List))
    <=> ( ( ( List = nil(B) )
         => aa(A,$o,P,F1) )
        & ( ( List = aa(list(B),list(B),cons(B,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List)) )
         => aa(A,$o,P,aa(list(B),A,aa(B,fun(list(B),A),F22,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ) ) ) ).

% list.split_sel
tff(fact_7647_list_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,fun(list(B),A)),List: list(B)] :
      ( aa(A,$o,P,aa(list(B),A,case_list(A,B,F1,F22),List))
    <=> ~ ( ( ( List = nil(B) )
            & ~ aa(A,$o,P,F1) )
          | ( ( List = aa(list(B),list(B),cons(B,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List)) )
            & ~ aa(A,$o,P,aa(list(B),A,aa(B,fun(list(B),A),F22,aa(list(B),B,hd(B),List)),aa(list(B),list(B),tl(B),List))) ) ) ) ).

% list.split_sel_asm
tff(fact_7648_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P: fun(nat,$o),Xs: list(A),Is: list(nat)] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_afq(fun(nat,$o),fun(product_prod(A,nat),$o),P)),zip(A,nat,Xs,Is))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_afr(fun(nat,$o),fun(product_prod(A,nat),$o),P)),zip(A,nat,Xs,aa(list(nat),list(nat),map(nat,nat,suc),Is)))) ).

% nths_shift_lemma_Suc
tff(fact_7649_take__Suc,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( Xs != nil(A) )
     => ( take(A,aa(nat,nat,suc,Nb),Xs) = aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),Xs)),take(A,Nb,aa(list(A),list(A),tl(A),Xs))) ) ) ).

% take_Suc
tff(fact_7650_rotate1__hd__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),rotate1(A),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xs)),aa(list(A),list(A),cons(A,aa(list(A),A,hd(A),Xs)),nil(A))) ) ) ).

% rotate1_hd_tl
tff(fact_7651_nths__shift__lemma,axiom,
    ! [A: $tType,A5: set(nat),Xs: list(A),I: nat] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_afs(set(nat),fun(product_prod(A,nat),$o),A5)),zip(A,nat,Xs,upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_aft(set(nat),fun(nat,fun(product_prod(A,nat),$o)),A5),I)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_7652_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] :
      aa(list(A),nat,size_list(A,F3),Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,aa(list(A),A,hd(A),Xs))),aa(list(A),nat,size_list(A,F3),aa(list(A),list(A),tl(A),Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_7653_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A5: set(nat)] : nths(A,Xs,A5) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_afs(set(nat),fun(product_prod(A,nat),$o),A5)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_7654_length__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = aa(nat,nat,foldr(nat,nat,times_times(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xss)),one_one(nat)) ).

% length_product_lists
tff(fact_7655_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P: fun(B,$o),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_afu(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F3),P),Xs) ).

% map_filter_map_filter
tff(fact_7656_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6)
     => ( aa(set(A),$o,finite_finite(A),X6)
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs)) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_afv(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F3)),X6) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_7657_map__upds__fold__map__upd,axiom,
    ! [A: $tType,B: $tType,Mb: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,Mb,Ks,Vs) = foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_afx(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Mb,zip(A,B,Ks,Vs)) ).

% map_upds_fold_map_upd
tff(fact_7658_foldl__append,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,A)),A2: A,Xs: list(B),Ys: list(B)] : foldl(A,B,F3,A2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = foldl(A,B,F3,foldl(A,B,F3,A2,Xs),Ys) ).

% foldl_append
tff(fact_7659_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( groups8242544230860333062m_list(A,Ns) = zero_zero(A) )
        <=> ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Ns))
             => ( X3 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_7660_sum__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xb: A,Xs: list(A)] : groups8242544230860333062m_list(A,aa(list(A),list(A),cons(A,Xb),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),groups8242544230860333062m_list(A,Xs)) ) ).

% sum_list.Cons
tff(fact_7661_sum__list__append,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A),Ys: list(A)] : groups8242544230860333062m_list(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs)),groups8242544230860333062m_list(A,Ys)) ) ).

% sum_list_append
tff(fact_7662_sum__list__upt,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb)
     => ( groups8242544230860333062m_list(nat,upt(Mb,Nb)) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ca(nat,nat)),set_or7035219750837199246ssThan(nat,Mb,Nb)) ) ) ).

% sum_list_upt
tff(fact_7663_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),groups8242544230860333062m_list(A,Xs))),groups8242544230860333062m_list(A,aa(list(A),list(A),map(A,A,abs_abs(A)),Xs))) ) ).

% sum_list_abs
tff(fact_7664_foldl__Cons,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,A)),A2: A,Xb: B,Xs: list(B)] : foldl(A,B,F3,A2,aa(list(B),list(B),cons(B,Xb),Xs)) = foldl(A,B,F3,aa(B,A,aa(A,fun(B,A),F3,A2),Xb),Xs) ).

% foldl_Cons
tff(fact_7665_foldl__Nil,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,A)),A2: A] : foldl(A,B,F3,A2,nil(B)) = A2 ).

% foldl_Nil
tff(fact_7666_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Xs: list(A)] :
          ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),groups8242544230860333062m_list(A,Xs)) ) ) ).

% member_le_sum_list
tff(fact_7667_foldl__cong,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: A,L: list(B),K2: list(B),F3: fun(A,fun(B,A)),G3: fun(A,fun(B,A))] :
      ( ( A2 = B2 )
     => ( ( L = K2 )
       => ( ! [A4: A,X4: B] :
              ( member(B,X4,aa(list(B),set(B),set2(B),L))
             => ( aa(B,A,aa(A,fun(B,A),F3,A4),X4) = aa(B,A,aa(A,fun(B,A),G3,A4),X4) ) )
         => ( foldl(A,B,F3,A2,L) = foldl(A,B,G3,B2,K2) ) ) ) ) ).

% foldl_cong
tff(fact_7668_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),zero_zero(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),groups8242544230860333062m_list(A,Xs)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_7669_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X4) )
         => ( ( groups8242544230860333062m_list(A,Xs) = zero_zero(A) )
          <=> ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
               => ( X3 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_7670_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X4) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),groups8242544230860333062m_list(A,Xs)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_7671_length__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,Xss)) = groups8242544230860333062m_list(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xss)) ).

% length_concat
tff(fact_7672_foldl__map,axiom,
    ! [A: $tType,B: $tType,C: $tType,G3: fun(A,fun(B,A)),A2: A,F3: fun(C,B),Xs: list(C)] : foldl(A,B,G3,A2,aa(list(C),list(B),map(C,B,F3),Xs)) = foldl(A,C,aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_afy(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G3),F3),A2,Xs) ).

% foldl_map
tff(fact_7673_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bd(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),Xs)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G3),Xs))) ) ).

% sum_list_subtractf
tff(fact_7674_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(B,A),G3: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bc(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G3)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G3),Xs))) ) ).

% sum_list_addf
tff(fact_7675_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C3: A,F3: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bb(A,fun(fun(B,A),fun(B,A)),C3),F3)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))) ) ).

% sum_list_const_mult
tff(fact_7676_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F3: fun(B,A),C3: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_ba(fun(B,A),fun(A,fun(B,A)),F3),C3)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))),C3) ) ).

% sum_list_mult_const
tff(fact_7677_sum__list__replicate,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat,C3: A] : groups8242544230860333062m_list(A,replicate(A,Nb,C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),C3) ) ).

% sum_list_replicate
tff(fact_7678_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = aa(A,A,foldr(A,A,plus_plus(A),Xs),zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_7679_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),aa(list(A),list(A),filter2(A,P),Xs)))),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))) ).

% sum_list_filter_le_nat
tff(fact_7680_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G3,X4)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs))) ) ) ).

% sum_list_mono
tff(fact_7681_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( groups8242544230860333062m_list(A,Xs) = aa(set(A),A,groups7311177749621191930dd_sum(A,A,aTP_Lamp_afz(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_7682_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X4)),aa(A,B,G3,X4)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs))) ) ) ) ).

% sum_list_strict_mono
tff(fact_7683_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K2: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K2)),groups8242544230860333062m_list(A,Ns)) ) ) ).

% elem_le_sum_list
tff(fact_7684_sum__list__map__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(A),P: fun(A,$o),F3: fun(A,B)] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => ( ~ aa(A,$o,P,X4)
               => ( aa(A,B,F3,X4) = zero_zero(B) ) ) )
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),filter2(A,P),Xs))) = groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_7685_sum_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(A),G3: fun(A,B)] :
          ( distinct(A,Xs)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G3),aa(list(A),set(A),set2(A),Xs)) = groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs)) ) ) ) ).

% sum.distinct_set_conv_list
tff(fact_7686_sum__list__distinct__conv__sum__set,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(A),F3: fun(A,B)] :
          ( distinct(A,Xs)
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% sum_list_distinct_conv_sum_set
tff(fact_7687_sum__list__map__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Xb: A,Xs: list(A),F3: fun(A,B)] :
          ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,Xb)),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),remove1(A,Xb,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_7688_sum__code,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),Xs: list(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G3),aa(list(B),set(B),set2(B),Xs)) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G3),remdups(B,Xs))) ) ).

% sum_code
tff(fact_7689_sum__set__upto__conv__sum__list__int,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(int,A),I: int,J: int] : aa(set(int),A,groups7311177749621191930dd_sum(int,A,F3),aa(list(int),set(int),set2(int),upto(I,J))) = groups8242544230860333062m_list(A,aa(list(int),list(A),map(int,A,F3),upto(I,J))) ) ).

% sum_set_upto_conv_sum_list_int
tff(fact_7690_interv__sum__list__conv__sum__set__int,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(int,A),K2: int,L: int] : groups8242544230860333062m_list(A,aa(list(int),list(A),map(int,A,F3),upto(K2,L))) = aa(set(int),A,groups7311177749621191930dd_sum(int,A,F3),aa(list(int),set(int),set2(int),upto(K2,L))) ) ).

% interv_sum_list_conv_sum_set_int
tff(fact_7691_size__list__conv__sum__list,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : aa(list(A),nat,size_list(A,F3),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% size_list_conv_sum_list
tff(fact_7692_sum__list__triv,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [R2: A,Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_kv(A,fun(B,A),R2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(list(B),nat,size_size(list(B)),Xs))),R2) ) ).

% sum_list_triv
tff(fact_7693_sum__list__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,aTP_Lamp_kz(fun(A,nat),fun(A,nat),F3)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_7694_sum__set__upt__conv__sum__list__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mb: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(list(nat),set(nat),set2(nat),upt(Mb,Nb))) = groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F3),upt(Mb,Nb))) ) ).

% sum_set_upt_conv_sum_list_nat
tff(fact_7695_interv__sum__list__conv__sum__set__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mb: nat,Nb: nat] : groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F3),upt(Mb,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(list(nat),set(nat),set2(nat),upt(Mb,Nb))) ) ).

% interv_sum_list_conv_sum_set_nat
tff(fact_7696_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_7697_card__length__sum__list__rec,axiom,
    ! [Mb: nat,N3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Mb)
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_aga(nat,fun(nat,fun(list(nat),$o)),Mb),N3))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_agb(nat,fun(nat,fun(list(nat),$o)),Mb),N3)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_agc(nat,fun(nat,fun(list(nat),$o)),Mb),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_7698_card__length__sum__list,axiom,
    ! [Mb: nat,N3: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_aga(nat,fun(nat,fun(list(nat),$o)),Mb),N3))) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),Mb)),one_one(nat))),N3) ).

% card_length_sum_list
tff(fact_7699_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs)) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_agd(fun(A,nat),fun(list(A),fun(A,nat)),F3),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_7700_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K2: nat,Xs: list(A),Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Xs))
         => ( groups8242544230860333062m_list(A,list_update(A,Xs,K2,Xb)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs)),Xb)),aa(nat,A,nth(A,Xs),K2)) ) ) ) ).

% sum_list_update
tff(fact_7701_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F3: fun(nat,A),Ns: list(nat)] :
          ( ! [X4: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X4),Y3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,X4)),aa(nat,A,F3,Y3)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F3),Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_7702_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A2: A,Xs: list(A),F3: fun(A,B)] :
          ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F3),Xs))
           => ( ( aa(list(A),A,hd(A),aa(list(A),list(A),filter2(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_age(A,fun(fun(A,B),fun(A,$o)),A2),F3)),Xs)) = A2 )
             => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,B,F3),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_7703_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),G3: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_agf(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F3),G3),Xs)),Xs))) ) ).

% sorted_map_same
tff(fact_7704_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P: fun(B,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs))) ) ) ).

% sorted_filter
tff(fact_7705_sorted__wrt__filter,axiom,
    ! [A: $tType,F3: fun(A,fun(A,$o)),Xs: list(A),P: fun(A,$o)] :
      ( sorted_wrt(A,F3,Xs)
     => sorted_wrt(A,F3,aa(list(A),list(A),filter2(A,P),Xs)) ) ).

% sorted_wrt_filter
tff(fact_7706_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G3: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_agg(fun(list(A),A),fun(list(A),fun(A,$o)),G3),Xs)),Xs)) ) ).

% sorted_same
tff(fact_7707_sorted__tl,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),tl(A),Xs)) ) ) ).

% sorted_tl
tff(fact_7708_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,$o)),F3: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R,aa(list(B),list(A),map(B,A,F3),Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_agh(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R),F3),Xs) ) ).

% sorted_wrt_map
tff(fact_7709_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),Xb: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),remove1(B,Xb,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_7710_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
        <=> sorted_wrt(B,aTP_Lamp_agi(fun(B,A),fun(B,fun(B,$o)),F3),Xs) ) ) ).

% sorted_map
tff(fact_7711_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xb: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ).

% sorted_insort_key
tff(fact_7712_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys)
             => ( distinct(A,Ys)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                 => ( Xs = Ys ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_7713_strict__sorted__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less(A),L)
        <=> ( sorted_wrt(A,ord_less_eq(A),L)
            & distinct(A,L) ) ) ) ).

% strict_sorted_iff
tff(fact_7714_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A),I: nat,J: nat] :
      ( sorted_wrt(A,P,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_7715_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_7716_sorted__wrt_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A,Ys: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),cons(A,Xb),Ys))
    <=> ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
           => aa(A,$o,aa(A,fun(A,$o),P,Xb),X3) )
        & sorted_wrt(A,P,Ys) ) ) ).

% sorted_wrt.simps(2)
tff(fact_7717_sorted__wrt_Oelims_I3_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( ~ sorted_wrt(A,Xb,Xaa)
     => ~ ! [X4: A,Ys3: list(A)] :
            ( ( Xaa = aa(list(A),list(A),cons(A,X4),Ys3) )
           => ( ! [Xa3: A] :
                  ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys3))
                 => aa(A,$o,aa(A,fun(A,$o),Xb,X4),Xa3) )
              & sorted_wrt(A,Xb,Ys3) ) ) ) ).

% sorted_wrt.elims(3)
tff(fact_7718_sorted__wrt__mono__rel,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,$o)),Q: fun(A,fun(A,$o))] :
      ( ! [X4: A,Y3: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => ( member(A,Y3,aa(list(A),set(A),set2(A),Xs))
           => ( aa(A,$o,aa(A,fun(A,$o),P,X4),Y3)
             => aa(A,$o,aa(A,fun(A,$o),Q,X4),Y3) ) ) )
     => ( sorted_wrt(A,P,Xs)
       => sorted_wrt(A,Q,Xs) ) ) ).

% sorted_wrt_mono_rel
tff(fact_7719_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys)
           => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_7720_sorted__wrt__append,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( sorted_wrt(A,P,Xs)
        & sorted_wrt(A,P,Ys)
        & ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
           => ! [Xa2: A] :
                ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),P,X3),Xa2) ) ) ) ) ).

% sorted_wrt_append
tff(fact_7721_sorted__wrt__take,axiom,
    ! [A: $tType,F3: fun(A,fun(A,$o)),Xs: list(A),Nb: nat] :
      ( sorted_wrt(A,F3,Xs)
     => sorted_wrt(A,F3,take(A,Nb,Xs)) ) ).

% sorted_wrt_take
tff(fact_7722_sorted__wrt__drop,axiom,
    ! [A: $tType,F3: fun(A,fun(A,$o)),Xs: list(A),Nb: nat] :
      ( sorted_wrt(A,F3,Xs)
     => sorted_wrt(A,F3,drop(A,Nb,Xs)) ) ).

% sorted_wrt_drop
tff(fact_7723_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),cons(A,Xb),Ys))
        <=> ( ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xb),X3) )
            & sorted_wrt(A,ord_less(A),Ys) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_7724_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => sorted_wrt(A,P,Xs) ) ).

% sorted_wrt01
tff(fact_7725_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
               => ! [Xa2: A] :
                    ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys))
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa2) ) ) ) ) ) ).

% sorted_append
tff(fact_7726_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Nb: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,Nb,Xs)) ) ) ).

% sorted_take
tff(fact_7727_sorted__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups(A,Xs)) ) ) ).

% sorted_remdups
tff(fact_7728_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Nb: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,Nb,Xs)) ) ) ).

% sorted_drop
tff(fact_7729_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Xb),Ys))
        <=> ( ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X3) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_7730_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Xb),nil(A))) ) ).

% sorted1
tff(fact_7731_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_7732_sorted__upto,axiom,
    ! [Mb: int,Nb: int] : sorted_wrt(int,ord_less_eq(int),upto(Mb,Nb)) ).

% sorted_upto
tff(fact_7733_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Nb: nat,Xb: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,Nb,Xb)) ) ).

% sorted_replicate
tff(fact_7734_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I6: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I6)) ) ) ).

% sorted_nths
tff(fact_7735_strict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% strict_sorted_imp_sorted
tff(fact_7736_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A2,Xs)) ) ) ).

% sorted_remove1
tff(fact_7737_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_7738_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),Xb),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_7739_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),cons(A,Y),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Y),Zs)) ) ) ) ).

% sorted2
tff(fact_7740_sorted__wrt1,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xb: A] : sorted_wrt(A,P,aa(list(A),list(A),cons(A,Xb),nil(A))) ).

% sorted_wrt1
tff(fact_7741_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_7742_sorted__wrt_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : sorted_wrt(A,P,nil(A)) ).

% sorted_wrt.simps(1)
tff(fact_7743_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_7744_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_7745_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_agj(A,fun(A,$o)),Xs) ).

% sorted_wrt_true
tff(fact_7746_sorted__remdups__adj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups_adj(A,Xs)) ) ) ).

% sorted_remdups_adj
tff(fact_7747_sorted__wrt__upt,axiom,
    ! [Mb: nat,Nb: nat] : sorted_wrt(nat,ord_less(nat),upt(Mb,Nb)) ).

% sorted_wrt_upt
tff(fact_7748_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_7749_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_7750_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_7751_sorted__wrt_Oelims_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A)] :
      ( sorted_wrt(A,Xb,Xaa)
     => ( ( Xaa != nil(A) )
       => ~ ! [X4: A,Ys3: list(A)] :
              ( ( Xaa = aa(list(A),list(A),cons(A,X4),Ys3) )
             => ~ ( ! [Xa: A] :
                      ( member(A,Xa,aa(list(A),set(A),set2(A),Ys3))
                     => aa(A,$o,aa(A,fun(A,$o),Xb,X4),Xa) )
                  & sorted_wrt(A,Xb,Ys3) ) ) ) ) ).

% sorted_wrt.elims(2)
tff(fact_7752_sorted__wrt_Oelims_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A),Y: $o] :
      ( ( sorted_wrt(A,Xb,Xaa)
      <=> (Y) )
     => ( ( ( Xaa = nil(A) )
         => ~ (Y) )
       => ~ ! [X4: A,Ys3: list(A)] :
              ( ( Xaa = aa(list(A),list(A),cons(A,X4),Ys3) )
             => ( (Y)
              <=> ~ ( ! [Xa2: A] :
                        ( member(A,Xa2,aa(list(A),set(A),set2(A),Ys3))
                       => aa(A,$o,aa(A,fun(A,$o),Xb,X4),Xa2) )
                    & sorted_wrt(A,Xb,Ys3) ) ) ) ) ) ).

% sorted_wrt.elims(1)
tff(fact_7753_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ? [X4: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X4) = A5 )
              & sorted_wrt(A,ord_less_eq(A),X4)
              & distinct(A,X4)
              & ! [Y4: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y4) = A5 )
                    & sorted_wrt(A,ord_less_eq(A),Y4)
                    & distinct(A,Y4) )
                 => ( Y4 = X4 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_7754_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_7755_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P: fun(B,$o),Xb: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => ( aa(B,$o,P,Xb)
           => ( aa(list(B),list(B),filter2(B,P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Xb),aa(list(B),list(B),filter2(B,P),Xs)) ) ) ) ) ).

% filter_insort
tff(fact_7756_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,Xs: list(A)] :
          ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_7757_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I3))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_7758_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ~ ! [L3: list(A)] :
                ( sorted_wrt(A,ord_less(A),L3)
               => ( ( aa(list(A),set(A),set2(A),L3) = A5 )
                 => ( aa(list(A),nat,size_size(list(A)),L3) != aa(set(A),nat,finite_card(A),A5) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_7759_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_7760_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_7761_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_7762_sorted__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Nb,Xs))) ).

% sorted_enumerate
tff(fact_7763_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A),Ys: list(A)] :
          ( inj_on(A,B,F3,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)))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F3),Xs))
           => ( distinct(B,aa(list(A),list(B),map(A,B,F3),Xs))
             => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F3),Ys))
               => ( distinct(B,aa(list(A),list(B),map(A,B,F3),Ys))
                 => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_7764_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),L: list(A)] :
          ( aa(set(A),$o,finite_finite(A),A5)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A5 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A5) ) )
          <=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A5) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_7765_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X4: A] :
                ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A2) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ms(A,A)),A2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,A2),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_7766_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afl(nat,fun(list(A),$o),I)),Xs)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_7767_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_afk(nat,fun(list(A),A),I)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afl(nat,fun(list(A),$o),I)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_7768_rev__is__rev__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),rev(A),Ys) )
    <=> ( Xs = Ys ) ) ).

% rev_is_rev_conv
tff(fact_7769_rev__rev__ident,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),rev(A),Xs)) = Xs ).

% rev_rev_ident
tff(fact_7770_rev__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rev_is_Nil_conv
tff(fact_7771_Nil__is__rev__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),rev(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% Nil_is_rev_conv
tff(fact_7772_set__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rev
tff(fact_7773_length__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rev
tff(fact_7774_rev__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),rev(A),Xs)) ).

% rev_append
tff(fact_7775_distinct__rev,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),rev(A),Xs))
    <=> distinct(A,Xs) ) ).

% distinct_rev
tff(fact_7776_rev__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),list(A),rev(A),replicate(A,Nb,Xb)) = replicate(A,Nb,Xb) ).

% rev_replicate
tff(fact_7777_remdups__adj__rev,axiom,
    ! [A: $tType,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),remdups_adj(A,Xs)) ).

% remdups_adj_rev
tff(fact_7778_inj__on__rev,axiom,
    ! [A: $tType,A5: set(list(A))] : inj_on(list(A),list(A),rev(A),A5) ).

% inj_on_rev
tff(fact_7779_singleton__rev__conv,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),cons(A,Xb),nil(A)) = aa(list(A),list(A),rev(A),Xs) )
    <=> ( aa(list(A),list(A),cons(A,Xb),nil(A)) = Xs ) ) ).

% singleton_rev_conv
tff(fact_7780_rev__singleton__conv,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),cons(A,Xb),nil(A)) )
    <=> ( Xs = aa(list(A),list(A),cons(A,Xb),nil(A)) ) ) ).

% rev_singleton_conv
tff(fact_7781_rev__eq__Cons__iff,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),cons(A,Y),Ys) )
    <=> ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),cons(A,Y),nil(A))) ) ) ).

% rev_eq_Cons_iff
tff(fact_7782_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xs: list(B),A2: A] : aa(A,A,foldr(B,A,F3,Xs),A2) = foldl(A,B,aTP_Lamp_agk(fun(B,fun(A,A)),fun(A,fun(B,A)),F3),A2,aa(list(B),list(B),rev(B),Xs)) ).

% foldr_conv_foldl
tff(fact_7783_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,A)),A2: A,Xs: list(B)] : foldl(A,B,F3,A2,Xs) = aa(A,A,foldr(B,A,aTP_Lamp_agl(fun(A,fun(B,A)),fun(B,fun(A,A)),F3),aa(list(B),list(B),rev(B),Xs)),A2) ).

% foldl_conv_foldr
tff(fact_7784_rev__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),list(A),rev(A),concat(A,Xs)) = concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),rev(A)),aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) ).

% rev_concat
tff(fact_7785_rev__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rev(A),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),rev(B),Xs)) ).

% rev_map
tff(fact_7786_sorted__wrt__rev,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),rev(A),Xs))
    <=> sorted_wrt(A,aTP_Lamp_agm(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P),Xs) ) ).

% sorted_wrt_rev
tff(fact_7787_rev__swap,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = Ys )
    <=> ( Xs = aa(list(A),list(A),rev(A),Ys) ) ) ).

% rev_swap
tff(fact_7788_rev_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rev(A),nil(A)) = nil(A) ).

% rev.simps(1)
tff(fact_7789_rev_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xs)),aa(list(A),list(A),cons(A,Xb),nil(A))) ).

% rev.simps(2)
tff(fact_7790_zip__rev,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( zip(A,B,aa(list(A),list(A),rev(A),Xs),aa(list(B),list(B),rev(B),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),rev(product_prod(A,B)),zip(A,B,Xs,Ys)) ) ) ).

% zip_rev
tff(fact_7791_rev__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),rev(A),Xs)) ).

% rev_filter
tff(fact_7792_take__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : take(A,Nb,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs)) ).

% take_rev
tff(fact_7793_rev__take,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),take(A,I,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I),aa(list(A),list(A),rev(A),Xs)) ).

% rev_take
tff(fact_7794_rev__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),drop(A,I,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),I),aa(list(A),list(A),rev(A),Xs)) ).

% rev_drop
tff(fact_7795_drop__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : drop(A,Nb,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb),Xs)) ).

% drop_rev
tff(fact_7796_rev__nth,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,aa(list(A),list(A),rev(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,Nb))) ) ) ).

% rev_nth
tff(fact_7797_rev__update,axiom,
    ! [A: $tType,K2: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),rev(A),list_update(A,Xs,K2,Y)) = list_update(A,aa(list(A),list(A),rev(A),Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K2)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_7798_sorted__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_7799_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I3))),aa(nat,A,nth(A,Xs),I3)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_7800_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I3)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_7801_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_7802_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Y) = $ite(Xs = nil(A),Y,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y)) ) ) ) ).

% foldr_max_sorted
tff(fact_7803_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = $ite(Xs = nil(list(A)),zero_zero(nat),aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_7804_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_afl(nat,fun(list(A),$o),I)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).

% transpose_column_length
tff(fact_7805_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S3)
       => ( aa(set(B),$o,finite_finite(B),A5)
         => ~ ! [L3: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L3))
               => ( ( aa(list(B),set(B),set2(B),L3) = A5 )
                 => ( aa(list(B),nat,size_size(list(B)),L3) != aa(set(B),nat,finite_card(B),A5) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_7806_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_aet(list(A),$o),Xs) ) ) ).

% transpose_transpose
tff(fact_7807_takeWhile__idem,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : takeWhile(A,P,takeWhile(A,P,Xs)) = takeWhile(A,P,Xs) ).

% takeWhile_idem
tff(fact_7808_takeWhile__eq__all__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( takeWhile(A,P,Xs) = Xs )
    <=> ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) ) ) ).

% takeWhile_eq_all_conv
tff(fact_7809_takeWhile__append1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = takeWhile(A,P,Xs) ) ) ) ).

% takeWhile_append1
tff(fact_7810_takeWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) )
     => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P,Ys)) ) ) ).

% takeWhile_append2
tff(fact_7811_takeWhile__replicate,axiom,
    ! [A: $tType,P: fun(A,$o),Nb: nat,Xb: A] :
      takeWhile(A,P,replicate(A,Nb,Xb)) = $ite(aa(A,$o,P,Xb),replicate(A,Nb,Xb),nil(A)) ).

% takeWhile_replicate
tff(fact_7812_length__concat__rev,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) = aa(list(A),nat,size_size(list(A)),concat(A,Xs)) ).

% length_concat_rev
tff(fact_7813_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_7814_zip__takeWhile__fst,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),Xs: list(A),Ys: list(B)] : zip(A,B,takeWhile(A,P,Xs),Ys) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),$o),comp(A,$o,product_prod(A,B),P),product_fst(A,B)),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_fst
tff(fact_7815_zip__takeWhile__snd,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),P: fun(B,$o),Ys: list(B)] : zip(A,B,Xs,takeWhile(B,P,Ys)) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),$o),comp(B,$o,product_prod(A,B),P),product_snd(A,B)),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_snd
tff(fact_7816_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))
     => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ).

% nth_length_takeWhile
tff(fact_7817_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,takeWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_7818_distinct__takeWhile,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,takeWhile(A,P,Xs)) ) ).

% distinct_takeWhile
tff(fact_7819_set__takeWhileD,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),takeWhile(A,P,Xs)))
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
        & aa(A,$o,P,Xb) ) ) ).

% set_takeWhileD
tff(fact_7820_takeWhile__cong,axiom,
    ! [A: $tType,L: list(A),K2: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( L = K2 )
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,P,X4)
            <=> aa(A,$o,Q,X4) ) )
       => ( takeWhile(A,P,L) = takeWhile(A,Q,K2) ) ) ) ).

% takeWhile_cong
tff(fact_7821_takeWhile__eq__take,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : takeWhile(A,P,Xs) = take(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)),Xs) ).

% takeWhile_eq_take
tff(fact_7822_takeWhile__eq__Nil__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( takeWhile(A,P,Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ~ aa(A,$o,P,aa(list(A),A,hd(A),Xs)) ) ) ).

% takeWhile_eq_Nil_iff
tff(fact_7823_length__takeWhile__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_takeWhile_le
tff(fact_7824_folding__insort__key_Oinj__on,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => inj_on(B,A,F3,S3) ) ).

% folding_insort_key.inj_on
tff(fact_7825_takeWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      takeWhile(A,P,aa(list(A),list(A),cons(A,Xb),Xs)) = $ite(aa(A,$o,P,Xb),aa(list(A),list(A),cons(A,Xb),takeWhile(A,P,Xs)),nil(A)) ).

% takeWhile.simps(2)
tff(fact_7826_takeWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : takeWhile(A,P,nil(A)) = nil(A) ).

% takeWhile.simps(1)
tff(fact_7827_takeWhile__tail,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A),L: list(A)] :
      ( ~ aa(A,$o,P,Xb)
     => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Xb),L))) = takeWhile(A,P,Xs) ) ) ).

% takeWhile_tail
tff(fact_7828_folding__insort__key_Odistinct__if__distinct__map,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
       => distinct(B,Xs) ) ) ).

% folding_insort_key.distinct_if_distinct_map
tff(fact_7829_takeWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: fun(B,A),Xs: list(B)] : takeWhile(A,P,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),takeWhile(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P),F3),Xs)) ).

% takeWhile_map
tff(fact_7830_takeWhile__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] :
      takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(
        ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) ),
        aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P,Ys)),
        takeWhile(A,P,Xs) ) ).

% takeWhile_append
tff(fact_7831_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_7832_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) ) )
     => ( ( aa(nat,$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)) )
       => ( takeWhile(A,P,Xs) = take(A,Nb,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_7833_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => folding_insort_key(A,A,ord_less_eq(A),ord_less(A),top_top(set(A)),aTP_Lamp_ms(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_7834_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),Ta: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),map(B,A,F3),Xs)))
         => ( aa(list(B),list(B),filter2(B,aa(A,fun(B,$o),aTP_Lamp_agn(fun(B,A),fun(A,fun(B,$o)),F3),Ta)),Xs) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_agn(fun(B,A),fun(A,fun(B,$o)),F3),Ta),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_7835_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),A5: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S3)
       => ( aa(set(B),$o,finite_finite(B),A5)
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L))
              & ( aa(list(B),set(B),set2(B),L) = A5 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A5) ) )
          <=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_7836_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),Xb: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),insert(B,Xb),A5)),S3)
       => ( aa(set(B),$o,finite_finite(B),A5)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),insert(B,Xb),bot_bot(set(B))))) = remove1(B,Xb,aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_7837_linorder_Osorted__key__list__of__set_Ocong,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(B,fun(B,$o))] : sorted8670434370408473282of_set(B,A,Less_eq) = sorted8670434370408473282of_set(B,A,Less_eq) ).

% linorder.sorted_key_list_of_set.cong
tff(fact_7838_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),A5: set(B),B5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S3)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B5),S3)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5) = aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),B5) )
           => ( aa(set(B),$o,finite_finite(B),A5)
             => ( aa(set(B),$o,finite_finite(B),B5)
               => ( A5 = B5 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_7839_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_7840_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S3)
       => ( aa(set(B),$o,finite_finite(B),A5)
         => ( aa(list(B),set(B),set2(B),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5)) = A5 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_7841_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S3)
       => ( aa(list(B),nat,size_size(list(B)),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5)) = aa(set(B),nat,finite_card(B),A5) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_7842_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S3)
       => distinct(A,aa(list(B),list(A),map(B,A,F3),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_7843_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S3)
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_7844_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S3)
       => sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_7845_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S3)
       => ( aa(set(B),$o,finite_finite(B),A5)
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5) = nil(B) )
          <=> ( A5 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7846_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S3)
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),Xs))
         => ( distinct(B,Xs)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_7847_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),Xb: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),insert(B,Xb),A5)),S3)
       => ( aa(set(B),$o,finite_finite(B),A5)
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),insert(B,Xb),A5)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Xb),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),insert(B,Xb),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_7848_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),Xb: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),insert(B,Xb),A5)),S3)
       => ( aa(set(B),$o,finite_finite(B),A5)
         => ( ~ member(B,Xb,A5)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),insert(B,Xb),A5)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Xb),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A5)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_7849_linorder_Oinsort__key_Ocong,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(B,fun(B,$o))] : insort_key(B,A,Less_eq) = insort_key(B,A,Less_eq) ).

% linorder.insort_key.cong
tff(fact_7850_folding__insort__key_Oinsort__key__commute,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S3: set(B),F3: fun(B,A),Xb: B,Y: B] :
      ( folding_insort_key(A,B,Less_eq,Less,S3,F3)
     => ( member(B,Xb,S3)
       => ( member(B,Y,S3)
         => ( aa(fun(list(B),list(B)),fun(list(B),list(B)),comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Y)),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Xb)) = aa(fun(list(B),list(B)),fun(list(B),list(B)),comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Xb)),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Y)) ) ) ) ) ).

% folding_insort_key.insort_key_commute
tff(fact_7851_extract__def,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : extract(A,P,Xs) = aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),case_list(option(product_prod(list(A),product_prod(A,list(A)))),A,none(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_ago(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs)),dropWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P),Xs)) ).

% extract_def
tff(fact_7852_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
                & aa(A,$o,P,X) )
           => ( find(A,P,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_agp(list(A),fun(fun(A,$o),fun(A,$o)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_7853_dropWhile__idem,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : dropWhile(A,P,dropWhile(A,P,Xs)) = dropWhile(A,P,Xs) ).

% dropWhile_idem
tff(fact_7854_dropWhile__eq__Nil__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) = nil(A) )
    <=> ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) ) ) ).

% dropWhile_eq_Nil_conv
tff(fact_7855_dropWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) )
     => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = dropWhile(A,P,Ys) ) ) ).

% dropWhile_append2
tff(fact_7856_dropWhile__append1,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xb)
       => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),Ys) ) ) ) ).

% dropWhile_append1
tff(fact_7857_dropWhile__replicate,axiom,
    ! [A: $tType,P: fun(A,$o),Nb: nat,Xb: A] :
      dropWhile(A,P,replicate(A,Nb,Xb)) = $ite(aa(A,$o,P,Xb),nil(A),replicate(A,Nb,Xb)) ).

% dropWhile_replicate
tff(fact_7858_takeWhile__dropWhile__id,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),takeWhile(A,P,Xs)),dropWhile(A,P,Xs)) = Xs ).

% takeWhile_dropWhile_id
tff(fact_7859_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu2: fun(A,$o)] : find(A,Uu2,nil(A)) = none(A) ).

% find.simps(1)
tff(fact_7860_remdups__adj__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(A),list(A),cons(A,Xb),remdups_adj(A,dropWhile(A,aTP_Lamp_agq(A,fun(A,$o),Xb),Xs))) ).

% remdups_adj_Cons'
tff(fact_7861_dropWhile__append3,axiom,
    ! [A: $tType,P: fun(A,$o),Y: A,Xs: list(A),Ys: list(A)] :
      ( ~ aa(A,$o,P,Y)
     => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),aa(list(A),list(A),cons(A,Y),Ys)) ) ) ).

% dropWhile_append3
tff(fact_7862_dropWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      dropWhile(A,P,aa(list(A),list(A),cons(A,Xb),Xs)) = $ite(aa(A,$o,P,Xb),dropWhile(A,P,Xs),aa(list(A),list(A),cons(A,Xb),Xs)) ).

% dropWhile.simps(2)
tff(fact_7863_dropWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : dropWhile(A,P,nil(A)) = nil(A) ).

% dropWhile.simps(1)
tff(fact_7864_length__dropWhile__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_dropWhile_le
tff(fact_7865_hd__dropWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) != nil(A) )
     => ~ aa(A,$o,P,aa(list(A),A,hd(A),dropWhile(A,P,Xs))) ) ).

% hd_dropWhile
tff(fact_7866_dropWhile__eq__self__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) = Xs )
    <=> ( ( Xs = nil(A) )
        | ~ aa(A,$o,P,aa(list(A),A,hd(A),Xs)) ) ) ).

% dropWhile_eq_self_iff
tff(fact_7867_set__dropWhileD,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),dropWhile(A,P,Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% set_dropWhileD
tff(fact_7868_dropWhile__cong,axiom,
    ! [A: $tType,L: list(A),K2: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( L = K2 )
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),L))
           => ( aa(A,$o,P,X4)
            <=> aa(A,$o,Q,X4) ) )
       => ( dropWhile(A,P,L) = dropWhile(A,Q,K2) ) ) ) ).

% dropWhile_cong
tff(fact_7869_find__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys )
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,P,X4)
            <=> aa(A,$o,Q,X4) ) )
       => ( find(A,P,Xs) = find(A,Q,Ys) ) ) ) ).

% find_cong
tff(fact_7870_distinct__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( distinct(A,Xs)
     => distinct(A,dropWhile(A,P,Xs)) ) ).

% distinct_dropWhile
tff(fact_7871_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),dropWhile(A,P,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_7872_find_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      find(A,P,aa(list(A),list(A),cons(A,Xb),Xs)) = $ite(aa(A,$o,P,Xb),aa(A,option(A),some(A),Xb),find(A,P,Xs)) ).

% find.simps(2)
tff(fact_7873_find__dropWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : find(A,P,Xs) = aa(list(A),option(A),case_list(option(A),A,none(A),aTP_Lamp_agr(A,fun(list(A),option(A)))),dropWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P),Xs)) ).

% find_dropWhile
tff(fact_7874_dropWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: fun(B,A),Xs: list(B)] : dropWhile(A,P,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),dropWhile(B,aa(fun(B,A),fun(B,$o),comp(A,$o,B,P),F3),Xs)) ).

% dropWhile_map
tff(fact_7875_find__None__iff2,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( none(A) = find(A,P,Xs) )
    <=> ~ ? [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X3) ) ) ).

% find_None_iff2
tff(fact_7876_find__None__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( find(A,P,Xs) = none(A) )
    <=> ~ ? [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X3) ) ) ).

% find_None_iff
tff(fact_7877_dropWhile__eq__Cons__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Y: A,Ys: list(A)] :
      ( ( dropWhile(A,P,Xs) = aa(list(A),list(A),cons(A,Y),Ys) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),takeWhile(A,P,Xs)),aa(list(A),list(A),cons(A,Y),Ys)) )
        & ~ aa(A,$o,P,Y) ) ) ).

% dropWhile_eq_Cons_conv
tff(fact_7878_takeWhile__eq__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),dropWhile(A,P,Xs)))
         => ~ aa(A,$o,P,X4) )
     => ( takeWhile(A,P,Xs) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ).

% takeWhile_eq_filter
tff(fact_7879_dropWhile__eq__drop,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : dropWhile(A,P,Xs) = drop(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)),Xs) ).

% dropWhile_eq_drop
tff(fact_7880_dropWhile__append,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A)] :
      dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = $ite(
        ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) ),
        dropWhile(A,P,Ys),
        aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),Ys) ) ).

% dropWhile_append
tff(fact_7881_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),cons(A,Y),nil(A))))),remdups_adj(A,dropWhile(A,aTP_Lamp_agq(A,fun(A,$o),Y),Ys))) ).

% remdups_adj_append_dropWhile
tff(fact_7882_tl__remdups__adj,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( ( Ys != nil(A) )
     => ( aa(list(A),list(A),tl(A),remdups_adj(A,Ys)) = remdups_adj(A,dropWhile(A,aTP_Lamp_ags(list(A),fun(A,$o),Ys),aa(list(A),list(A),tl(A),Ys))) ) ) ).

% tl_remdups_adj
tff(fact_7883_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,dropWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_7884_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( dropWhile(A,aTP_Lamp_aff(A,fun(A,$o),Xb),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),cons(A,Xb),aa(list(A),list(A),rev(A),takeWhile(A,aTP_Lamp_aff(A,fun(A,$o),Xb),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_7885_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( distinct(A,Xs)
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => ( takeWhile(A,aTP_Lamp_aff(A,fun(A,$o),Xb),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),dropWhile(A,aTP_Lamp_aff(A,fun(A,$o),Xb),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_7886_find__Some__iff2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),Xb) = find(A,P,Xs) )
    <=> ? [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,$o,P,aa(nat,A,nth(A,Xs),I3))
          & ( Xb = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_7887_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Xb: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),Xb) )
    <=> ? [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,$o,P,aa(nat,A,nth(A,Xs),I3))
          & ( Xb = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_7888_partition__filter__conv,axiom,
    ! [A: $tType,F3: fun(A,$o),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,F3),Xs) = 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)),aa(list(A),list(A),filter2(A,F3),Xs)),aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),F3)),Xs)) ).

% partition_filter_conv
tff(fact_7889_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F3: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),aTP_Lamp_agt(fun(B,list(A)),fun(B,set(A)),F3),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_7890_bind__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,list(A))] : bind(B,A,nil(B),F3) = nil(A) ).

% bind_simps(1)
tff(fact_7891_bind__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xb: B,Xs: list(B),F3: fun(B,list(A))] : bind(B,A,aa(list(B),list(B),cons(B,Xb),Xs),F3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F3,Xb)),bind(B,A,Xs,F3)) ).

% bind_simps(2)
tff(fact_7892_partition__filter1,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ).

% partition_filter1
tff(fact_7893_list__bind__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F3: fun(A,list(B)),G3: fun(A,list(B))] :
      ( ( Xs = Ys )
     => ( ! [X4: A] :
            ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
           => ( aa(A,list(B),F3,X4) = aa(A,list(B),G3,X4) ) )
       => ( bind(A,B,Xs,F3) = bind(A,B,Ys,G3) ) ) ) ).

% list_bind_cong
tff(fact_7894_partition_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),nil(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)),nil(A)),nil(A)) ).

% partition.simps(1)
tff(fact_7895_partition__P,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = 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)),Yes),No4) )
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Yes))
           => aa(A,$o,P,X) )
        & ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),No4))
           => ~ aa(A,$o,P,X) ) ) ) ).

% partition_P
tff(fact_7896_partition_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),aa(list(A),list(A),cons(A,Xb),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(fun(list(A),fun(list(A),product_prod(list(A),list(A)))),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),product_case_prod(list(A),list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_agu(fun(A,$o),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P),Xb)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) ).

% partition.simps(2)
tff(fact_7897_partition__filter2,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),P)),Xs) ).

% partition_filter2
tff(fact_7898_List_Obind__def,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F3: fun(B,list(A))] : bind(B,A,Xs,F3) = concat(A,aa(list(B),list(list(A)),map(B,list(A),F3),Xs)) ).

% List.bind_def
tff(fact_7899_partition__set,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = 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)),Yes),No4) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes)),aa(list(A),set(A),set2(A),No4)) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% partition_set
tff(fact_7900_has__derivative__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(B,A),Xb: B,F6: fun(B,A),S3: set(B),Nb: int] :
          ( ( aa(B,A,F3,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,F3,F6,topolo174197925503356063within(B,Xb,S3))
           => has_derivative(B,A,aa(int,fun(B,A),aTP_Lamp_agv(fun(B,A),fun(int,fun(B,A)),F3),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_agw(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),F3),Xb),F6),Nb),topolo174197925503356063within(B,Xb,S3)) ) ) ) ).

% has_derivative_power_int
tff(fact_7901_has__derivative__power__int_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,Nb: int,S3: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_derivative(A,A,aTP_Lamp_agx(int,fun(A,A),Nb),aa(int,fun(A,A),aTP_Lamp_agy(A,fun(int,fun(A,A)),Xb),Nb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_power_int'
tff(fact_7902_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_7903_power__int__1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y: A] : power_int(A,Y,one_one(int)) = Y ) ).

% power_int_1_right
tff(fact_7904_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_7905_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_7906_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_7907_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_7908_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_7909_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_7910_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_7911_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_7912_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) = 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_7913_of__real__eq__numeral__power__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Y: real,Xb: num,Nb: int] :
          ( ( 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_7914_power__int__minus1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y: A] : power_int(A,Y,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Y) ) ).

% power_int_minus1_right
tff(fact_7915_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_7916_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_7917_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_7918_power__int__minus__left__even,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int,A2: A] :
          ( 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_7919_power__int__minus__left__odd,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Nb: int,A2: A] :
          ( ~ 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_7920_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_7921_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_7922_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_7923_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)) = divide_divide(A,power_int(A,Xb,Mb),power_int(A,Xb,Nb)) ) ) ) ).

% power_int_diff
tff(fact_7924_power__int__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,Nb: int] : power_int(A,divide_divide(A,one_one(A),Xb),Nb) = divide_divide(A,one_one(A),power_int(A,Xb,Nb)) ) ).

% power_int_one_over
tff(fact_7925_power__int__divide__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Mb: int] : power_int(A,divide_divide(A,Xb,Y),Mb) = divide_divide(A,power_int(A,Xb,Mb),power_int(A,Y,Mb)) ) ).

% power_int_divide_distrib
tff(fact_7926_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_7927_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_7928_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_7929_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_7930_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_7931_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N3: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N3)
         => ( aa(A,$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,N3)) ) ) ) ).

% power_int_strict_increasing
tff(fact_7932_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_7933_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N3: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N3)
         => ( aa(A,$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,N3)) ) ) ) ).

% power_int_increasing
tff(fact_7934_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N3: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Nb),N3)
         => ( aa(A,$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,N3)),power_int(A,A2,Nb)) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_7935_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_7936_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_7937_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_7938_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_7939_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_7940_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_7941_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_7942_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_7943_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_7944_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: int,N3: int,A2: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),N3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => ( ( ( A2 != zero_zero(A) )
                  | ( N3 != zero_zero(int) )
                  | ( Nb = zero_zero(int) ) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A2,N3)),power_int(A,A2,Nb)) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_7945_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_7946_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_7947_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(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_7948_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_7949_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_7950_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_7951_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_7952_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D3: A,Xb: A,S2: set(A),Nb: int] :
          ( has_field_derivative(A,F3,D3,topolo174197925503356063within(A,Xb,S2))
         => ( ( aa(A,A,F3,Xb) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_agz(fun(A,A),fun(int,fun(A,A)),F3),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Nb)),power_int(A,aa(A,A,F3,Xb),aa(int,int,aa(int,fun(int,int),minus_minus(int),Nb),one_one(int))))),D3),topolo174197925503356063within(A,Xb,S2)) ) ) ) ).

% DERIV_power_int
tff(fact_7953_lists__length__Suc__eq,axiom,
    ! [A: $tType,A5: set(A),Nb: nat] : aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_aha(set(A),fun(nat,fun(list(A),$o)),A5),Nb)) = 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_acp(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_af(set(A),fun(nat,fun(list(A),$o)),A5),Nb)),aTP_Lamp_ahb(set(A),fun(list(A),set(A)),A5))) ).

% lists_length_Suc_eq
tff(fact_7954_pred__nat__def,axiom,
    pred_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_ahc(nat,fun(nat,$o)))) ).

% pred_nat_def
tff(fact_7955_SigmaI,axiom,
    ! [B: $tType,A: $tType,A2: A,A5: set(A),B2: B,B5: fun(A,set(B))] :
      ( member(A,A2,A5)
     => ( member(B,B2,aa(A,set(B),B5,A2))
       => 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,A5,B5)) ) ) ).

% SigmaI
tff(fact_7956_mem__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A5: set(A),B5: fun(A,set(B))] :
      ( 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,A5,B5))
    <=> ( member(A,A2,A5)
        & member(B,B2,aa(A,set(B),B5,A2)) ) ) ).

% mem_Sigma_iff
tff(fact_7957_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,$o)),aTP_Lamp_ahd(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),P),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P),aTP_Lamp_ahe(fun(B,$o),fun(A,set(B)),Q)) ).

% Collect_case_prod
tff(fact_7958_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B5: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B5) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty1
tff(fact_7959_Compl__Times__UNIV1,axiom,
    ! [A: $tType,B: $tType,A5: 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_ahf(set(B),fun(A,set(B)),A5))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ahg(set(B),fun(A,set(B)),A5)) ).

% Compl_Times_UNIV1
tff(fact_7960_Compl__Times__UNIV2,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_ahh(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A5),aTP_Lamp_ahh(A,set(B))) ).

% Compl_Times_UNIV2
tff(fact_7961_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : product_Sigma(A,B,A5,aTP_Lamp_ahi(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_7962_Times__empty,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( ( product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A5 = bot_bot(set(A)) )
        | ( B5 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_7963_UNIV__Times__UNIV,axiom,
    ! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ahh(A,set(B))) = top_top(set(product_prod(A,B))) ).

% UNIV_Times_UNIV
tff(fact_7964_fst__image__times,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
      image(product_prod(A,B),A,product_fst(A,B),product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5))) = $ite(B5 = bot_bot(set(B)),bot_bot(set(A)),A5) ).

% fst_image_times
tff(fact_7965_snd__image__times,axiom,
    ! [B: $tType,A: $tType,A5: set(B),B5: set(A)] :
      image(product_prod(B,A),A,product_snd(B,A),product_Sigma(B,A,A5,aTP_Lamp_ahj(set(A),fun(B,set(A)),B5))) = $ite(A5 = bot_bot(set(B)),bot_bot(set(A)),B5) ).

% snd_image_times
tff(fact_7966_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_ahk(list(B),fun(A,set(B)),Ys)) ).

% set_product
tff(fact_7967_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A2: A,A5: set(A),B2: B,B5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),insert(A,A2),A5),aa(set(B),fun(A,set(B)),aTP_Lamp_ahl(B,fun(set(B),fun(A,set(B))),B2),B5)) = 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,A5,aa(set(B),fun(A,set(B)),aTP_Lamp_ahl(B,fun(set(B),fun(A,set(B))),B2),B5))),product_Sigma(A,B,aa(set(A),set(A),insert(A,A2),A5),aTP_Lamp_ahf(set(B),fun(A,set(B)),B5)))) ).

% insert_Times_insert
tff(fact_7968_inj__on__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C),A5: set(A)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F3),product_Sigma(A,B,A5,aTP_Lamp_ahh(A,set(B))))
    <=> inj_on(A,C,F3,A5) ) ).

% inj_on_apfst
tff(fact_7969_inj__on__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C),A5: 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),F3),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ahf(set(B),fun(A,set(B)),A5)))
    <=> inj_on(B,C,F3,A5) ) ).

% inj_on_apsnd
tff(fact_7970_UN__Times__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,E5: fun(C,set(A)),F4: fun(D,set(B)),A5: set(C),B5: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(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_ahn(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E5),F4)),product_Sigma(C,D,A5,aTP_Lamp_aho(set(D),fun(C,set(D)),B5)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(C,set(A),E5,A5)),aa(set(D),fun(A,set(B)),aTP_Lamp_ahp(fun(D,set(B)),fun(set(D),fun(A,set(B))),F4),B5)) ).

% UN_Times_distrib
tff(fact_7971_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] : image(product_prod(A,B),A,product_fst(A,B),product_Sigma(A,B,A5,B5)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_ahq(set(A),fun(fun(A,set(B)),fun(A,$o)),A5),B5)) ).

% fst_image_Sigma
tff(fact_7972_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),A5: set(C),B5: set(D)] : 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_adh(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3)),product_Sigma(C,D,A5,aTP_Lamp_aho(set(D),fun(C,set(D)),B5))) = product_Sigma(A,B,image(C,A,F3,A5),aa(set(D),fun(A,set(B)),aTP_Lamp_ahr(fun(D,B),fun(set(D),fun(A,set(B))),G3),B5)) ).

% image_paired_Times
tff(fact_7973_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,$o)),aTP_Lamp_ahs(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),P),Q))) = product_Sigma(A,B,aa(fun(A,$o),set(A),collect(A),P),aTP_Lamp_aht(fun(A,fun(B,$o)),fun(A,set(B)),Q)) ).

% Collect_case_prod_Sigma
tff(fact_7974_mem__Times__iff,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),A5: set(A),B5: set(B)] :
      ( member(product_prod(A,B),Xb,product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5)))
    <=> ( member(A,aa(product_prod(A,B),A,product_fst(A,B),Xb),A5)
        & member(B,aa(product_prod(A,B),B,product_snd(A,B),Xb),B5) ) ) ).

% mem_Times_iff
tff(fact_7975_swap__product,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B5: set(A)] : 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_mc(B,fun(A,product_prod(A,B)))),product_Sigma(B,A,A5,aTP_Lamp_ahj(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_ahf(set(B),fun(A,set(B)),A5)) ).

% swap_product
tff(fact_7976_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),A5: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_rtrancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_ahu(set(A),fun(A,set(A)),A5)))
       => ( ( A2 = B2 )
          | member(A,A2,A5) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_7977_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C5: set(A),B5: fun(A,set(B)),D4: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C5)
     => ( ! [X4: A] :
            ( member(A,X4,A5)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B5,X4)),aa(A,set(B),D4,X4)) )
       => aa(set(product_prod(A,B)),$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,A5,B5)),product_Sigma(A,B,C5,D4)) ) ) ).

% Sigma_mono
tff(fact_7978_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,Xb: A,C5: set(A),A5: set(B),B5: set(B)] :
      ( member(A,Xb,C5)
     => ( aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A5,aTP_Lamp_ahj(set(A),fun(B,set(A)),C5))),product_Sigma(B,A,B5,aTP_Lamp_ahj(set(A),fun(B,set(A)),C5)))
      <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),B5) ) ) ).

% Times_subset_cancel2
tff(fact_7979_Times__Int__Times,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),C5: set(A),D4: 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,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5))),product_Sigma(A,B,C5,aTP_Lamp_ahf(set(B),fun(A,set(B)),D4))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C5),aa(set(B),fun(A,set(B)),aTP_Lamp_ahv(set(B),fun(set(B),fun(A,set(B))),B5),D4)) ).

% Times_Int_Times
tff(fact_7980_Sigma__Int__distrib2,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I6,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ahw(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A5),B5)) = 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,I6,A5)),product_Sigma(A,B,I6,B5)) ).

% Sigma_Int_distrib2
tff(fact_7981_Times__Int__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: 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)),A5),B5),aTP_Lamp_ahf(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,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_ahf(set(B),fun(A,set(B)),C5))) ).

% Times_Int_distrib1
tff(fact_7982_Sigma__Int__distrib1,axiom,
    ! [B: $tType,A: $tType,I6: 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)),inf_inf(set(A)),I6),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))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I6,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Int_distrib1
tff(fact_7983_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,I6: 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)),I6),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,I6,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Diff_distrib1
tff(fact_7984_Sigma__Diff__distrib2,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I6,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ahx(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A5),B5)) = 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,I6,A5)),product_Sigma(A,B,I6,B5)) ).

% Sigma_Diff_distrib2
tff(fact_7985_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: 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)),A5),B5),aTP_Lamp_ahf(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,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_ahf(set(B),fun(A,set(B)),C5))) ).

% Times_Diff_distrib1
tff(fact_7986_Sigma__cong,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(A),C5: fun(A,set(B)),D4: fun(A,set(B))] :
      ( ( A5 = B5 )
     => ( ! [X4: A] :
            ( member(A,X4,B5)
           => ( aa(A,set(B),C5,X4) = aa(A,set(B),D4,X4) ) )
       => ( product_Sigma(A,B,A5,C5) = product_Sigma(A,B,B5,D4) ) ) ) ).

% Sigma_cong
tff(fact_7987_Times__eq__cancel2,axiom,
    ! [A: $tType,B: $tType,Xb: A,C5: set(A),A5: set(B),B5: set(B)] :
      ( member(A,Xb,C5)
     => ( ( product_Sigma(B,A,A5,aTP_Lamp_ahj(set(A),fun(B,set(A)),C5)) = product_Sigma(B,A,B5,aTP_Lamp_ahj(set(A),fun(B,set(A)),C5)) )
      <=> ( A5 = B5 ) ) ) ).

% Times_eq_cancel2
tff(fact_7988_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I6: set(A),X6: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I6,X6) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X3: A] :
          ( member(A,X3,I6)
         => ( aa(A,set(B),X6,X3) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_7989_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),C5: set(A),D4: set(B)] :
      ( ( product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5)) = product_Sigma(A,B,C5,aTP_Lamp_ahf(set(B),fun(A,set(B)),D4)) )
    <=> ( ( ( A5 = C5 )
          & ( B5 = D4 ) )
        | ( ( ( A5 = bot_bot(set(A)) )
            | ( B5 = bot_bot(set(B)) ) )
          & ( ( C5 = bot_bot(set(A)) )
            | ( D4 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_7990_SigmaE,axiom,
    ! [A: $tType,B: $tType,C3: product_prod(A,B),A5: set(A),B5: fun(A,set(B))] :
      ( member(product_prod(A,B),C3,product_Sigma(A,B,A5,B5))
     => ~ ! [X4: A] :
            ( member(A,X4,A5)
           => ! [Y3: B] :
                ( member(B,Y3,aa(A,set(B),B5,X4))
               => ( C3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) ) ) ) ) ).

% SigmaE
tff(fact_7991_SigmaD1,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A5: set(A),B5: fun(A,set(B))] :
      ( 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,A5,B5))
     => member(A,A2,A5) ) ).

% SigmaD1
tff(fact_7992_SigmaD2,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A5: set(A),B5: fun(A,set(B))] :
      ( 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,A5,B5))
     => member(B,B2,aa(A,set(B),B5,A2)) ) ).

% SigmaD2
tff(fact_7993_SigmaE2,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,A5: set(A),B5: fun(A,set(B))] :
      ( 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,A5,B5))
     => ~ ( member(A,A2,A5)
         => ~ member(B,B2,aa(A,set(B),B5,A2)) ) ) ).

% SigmaE2
tff(fact_7994_Times__Un__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: 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)),A5),B5),aTP_Lamp_ahf(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,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_ahf(set(B),fun(A,set(B)),C5))) ).

% Times_Un_distrib1
tff(fact_7995_Sigma__Un__distrib2,axiom,
    ! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I6,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ahy(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A5),B5)) = 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,I6,A5)),product_Sigma(A,B,I6,B5)) ).

% Sigma_Un_distrib2
tff(fact_7996_Sigma__Un__distrib1,axiom,
    ! [B: $tType,A: $tType,I6: 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)),sup_sup(set(A)),I6),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))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I6,C5)),product_Sigma(A,B,J4,C5)) ).

% Sigma_Un_distrib1
tff(fact_7997_member__product,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),A5: set(A),B5: set(B)] :
      ( member(product_prod(A,B),Xb,product_product(A,B,A5,B5))
    <=> member(product_prod(A,B),Xb,product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5))) ) ).

% member_product
tff(fact_7998_Product__Type_Oproduct__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] : product_product(A,B,A5,B5) = product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5)) ).

% Product_Type.product_def
tff(fact_7999_Sigma__Union,axiom,
    ! [B: $tType,A: $tType,X6: set(set(A)),B5: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X6),B5) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image(set(A),set(product_prod(A,B)),aTP_Lamp_ahz(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B5),X6)) ).

% Sigma_Union
tff(fact_8000_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),C5: set(B),B5: set(A),D4: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_ahf(set(B),fun(A,set(B)),D4)))
    <=> ( ( A5 = bot_bot(set(A)) )
        | ( C5 = bot_bot(set(B)) )
        | ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C5),D4) ) ) ) ).

% times_subset_iff
tff(fact_8001_card__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)) ).

% card_cartesian_product
tff(fact_8002_snd__image__Sigma,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B5: fun(B,set(A))] : image(product_prod(B,A),A,product_snd(B,A),product_Sigma(B,A,A5,B5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),B5,A5)) ).

% snd_image_Sigma
tff(fact_8003_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A5: set(product_prod(A,B))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A5),product_Sigma(A,B,image(product_prod(A,B),A,product_fst(A,B),A5),aTP_Lamp_aia(set(product_prod(A,B)),fun(A,set(B)),A5))) ).

% subset_fst_snd
tff(fact_8004_Sigma__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] : product_Sigma(A,B,A5,B5) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image(A,set(product_prod(A,B)),aTP_Lamp_aic(fun(A,set(B)),fun(A,set(product_prod(A,B))),B5),A5)) ).

% Sigma_def
tff(fact_8005_product__fold,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(B),$o,finite_finite(B),B5)
       => ( product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_aie(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B5),bot_bot(set(product_prod(A,B))),A5) ) ) ) ).

% product_fold
tff(fact_8006_less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mb),Nb),transitive_trancl(nat,pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mb),Nb) ) ).

% less_eq
tff(fact_8007_pred__nat__trancl__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mb),Nb),transitive_rtrancl(nat,pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mb),Nb) ) ).

% pred_nat_trancl_eq_le
tff(fact_8008_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_0(C)
     => ! [F3: fun(A,C),G3: fun(B,C),A5: set(A),B5: set(B)] :
          ( inj_on(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_aif(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),F3),G3)),product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5)))
         => ( aa(C,C,aa(C,fun(C,C),times_times(C),aa(set(A),C,groups7311177749621191930dd_sum(A,C,F3),A5)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G3),B5)) = aa(set(C),C,groups7311177749621191930dd_sum(C,C,id(C)),aa(fun(C,$o),set(C),collect(C),aa(set(B),fun(C,$o),aa(set(A),fun(set(B),fun(C,$o)),aa(fun(B,C),fun(set(A),fun(set(B),fun(C,$o))),aTP_Lamp_aig(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),F3),G3),A5),B5))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_8009_minus__fold__remove,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite(A),A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5) = finite_fold(A,set(A),remove(A),B5,A5) ) ) ).

% minus_fold_remove
tff(fact_8010_listrel__def,axiom,
    ! [B: $tType,A: $tType,X: set(product_prod(A,B))] : listrel(A,B,X) = aa(fun(product_prod(list(A),list(B)),$o),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),$o)),fun(product_prod(list(A),list(B)),$o),product_case_prod(list(A),list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),X)))) ).

% listrel_def
tff(fact_8011_listrelp_OCons,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Xb: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,Xb),Y)
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs),Ys)
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(B),list(B),cons(B,Y),Ys)) ) ) ).

% listrelp.Cons
tff(fact_8012_listrelp_ONil,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] : aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),nil(A)),nil(B)) ).

% listrelp.Nil
tff(fact_8013_listrelp_Ocases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: list(A),A22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),A1),A22)
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X4: A,Y3: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),cons(A,X4),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),cons(B,Y3),Ys3) )
                 => ( aa(B,$o,aa(A,fun(B,$o),R2,X4),Y3)
                   => ~ aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs2),Ys3) ) ) ) ) ) ).

% listrelp.cases
tff(fact_8014_listrelp_Osimps,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A1: list(A),A22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),A1),A22)
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X3: A,Y5: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),cons(A,X3),Xs3) )
            & ( A22 = aa(list(B),list(B),cons(B,Y5),Ys4) )
            & aa(B,$o,aa(A,fun(B,$o),R2,X3),Y5)
            & aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,R2),Xs3),Ys4) ) ) ) ).

% listrelp.simps
tff(fact_8015_product__atMost__eq__Un,axiom,
    ! [A5: set(nat),Mb: nat] : product_Sigma(nat,nat,A5,aTP_Lamp_aih(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,A5,aTP_Lamp_aii(nat,fun(nat,set(nat)),Mb))),product_Sigma(nat,nat,A5,aTP_Lamp_aij(nat,fun(nat,set(nat)),Mb))) ).

% product_atMost_eq_Un
tff(fact_8016_remove__code_I1_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xb),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,Xb),Xs)) ).

% remove_code(1)
tff(fact_8017_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_gi(nat,fun(nat,fun(nat,$o)),Mb))) = product_Sigma(nat,nat,set_ord_atMost(nat,Mb),aTP_Lamp_aii(nat,fun(nat,set(nat)),Mb)) ).

% pairs_le_eq_Sigma
tff(fact_8018_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),X: list(A),Xa: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),R2)),X),Xa)
    <=> member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),X),Xa),listrel(A,B,R2)) ) ).

% listrelp_listrel_eq
tff(fact_8019_remove__def,axiom,
    ! [A: $tType,Xb: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xb),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) ).

% remove_def
tff(fact_8020_mult__inj__if__coprime__nat,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),A5: set(A),G3: fun(B,nat),B5: set(B)] :
      ( inj_on(A,nat,F3,A5)
     => ( inj_on(B,nat,G3,B5)
       => ( ! [A4: A,B4: B] :
              ( member(A,A4,A5)
             => ( member(B,B4,B5)
               => algebr8660921524188924756oprime(nat,aa(A,nat,F3,A4),aa(B,nat,G3,B4)) ) )
         => inj_on(product_prod(A,B),nat,aa(fun(A,fun(B,nat)),fun(product_prod(A,B),nat),product_case_prod(A,B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_aik(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F3),G3)),product_Sigma(A,B,A5,aTP_Lamp_ahf(set(B),fun(A,set(B)),B5))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_8021_rotate__drop__take,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Nb),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)),Xs)),take(A,modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)),Xs)) ).

% rotate_drop_take
tff(fact_8022_coprime__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C3: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3)
        <=> ( algebr8660921524188924756oprime(A,A2,C3)
            & algebr8660921524188924756oprime(A,B2,C3) ) ) ) ).

% coprime_mult_left_iff
tff(fact_8023_coprime__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C3: A,A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C3,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( algebr8660921524188924756oprime(A,C3,A2)
            & algebr8660921524188924756oprime(A,C3,B2) ) ) ) ).

% coprime_mult_right_iff
tff(fact_8024_rotate__is__Nil__conv,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( aa(list(A),list(A),rotate(A,Nb),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rotate_is_Nil_conv
tff(fact_8025_set__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rotate
tff(fact_8026_length__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate
tff(fact_8027_distinct__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),rotate(A,Nb),Xs))
    <=> distinct(A,Xs) ) ).

% distinct_rotate
tff(fact_8028_coprime__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,A2,A2)
        <=> dvd_dvd(A,A2,one_one(A)) ) ) ).

% coprime_self
tff(fact_8029_coprime__mod__right__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,A2,modulo_modulo(A,B2,A2))
          <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mod_right_iff
tff(fact_8030_coprime__mod__left__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,modulo_modulo(A,A2,B2),B2)
          <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mod_left_iff
tff(fact_8031_coprime__power__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( algebr8660921524188924756oprime(A,A2,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
        <=> ( algebr8660921524188924756oprime(A,A2,B2)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_8032_coprime__power__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),B2)
        <=> ( algebr8660921524188924756oprime(A,A2,B2)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_8033_coprime__imp__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
         => ( gcd_gcd(A,A2,B2) = one_one(A) ) ) ) ).

% coprime_imp_gcd_eq_1
tff(fact_8034_rotate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,aa(nat,nat,suc,Nb)),Xs) = aa(list(A),list(A),rotate1(A),aa(list(A),list(A),rotate(A,Nb),Xs)) ).

% rotate_Suc
tff(fact_8035_rotate0,axiom,
    ! [A: $tType] : rotate(A,zero_zero(nat)) = id(list(A)) ).

% rotate0
tff(fact_8036_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,A2,zero_zero(A))
        <=> dvd_dvd(A,A2,one_one(A)) ) ) ).

% coprime_0_right_iff
tff(fact_8037_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A2)
        <=> dvd_dvd(A,A2,one_one(A)) ) ) ).

% coprime_0_left_iff
tff(fact_8038_coprime__mult__self__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( dvd_dvd(A,C3,one_one(A))
            & algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mult_self_left_iff
tff(fact_8039_coprime__mult__self__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( dvd_dvd(A,C3,one_one(A))
            & algebr8660921524188924756oprime(A,A2,B2) ) ) ) ).

% coprime_mult_self_right_iff
tff(fact_8040_is__unit__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,gcd_gcd(A,A2,B2),one_one(A))
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% is_unit_gcd
tff(fact_8041_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( aa(list(A),list(A),rotate(A,Nb),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_8042_rotate__id,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
     => ( aa(list(A),list(A),rotate(A,Nb),Xs) = Xs ) ) ).

% rotate_id
tff(fact_8043_coprime__right__2__iff__odd,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ).

% coprime_right_2_iff_odd
tff(fact_8044_coprime__left__2__iff__odd,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( algebr8660921524188924756oprime(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)
        <=> ~ dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2) ) ) ).

% coprime_left_2_iff_odd
tff(fact_8045_coprime__diff__one__left,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),A2) ) ).

% coprime_diff_one_left
tff(fact_8046_coprime__doff__one__right,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,A2,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))) ) ).

% coprime_doff_one_right
tff(fact_8047_rotate__add,axiom,
    ! [A: $tType,Mb: nat,Nb: nat] : rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),rotate(A,Mb)),rotate(A,Nb)) ).

% rotate_add
tff(fact_8048_rotate__append,axiom,
    ! [A: $tType,L: list(A),Q3: list(A)] : aa(list(A),list(A),rotate(A,aa(list(A),nat,size_size(list(A)),L)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),Q3)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Q3),L) ).

% rotate_append
tff(fact_8049_rotate__conv__mod,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Nb),Xs) = aa(list(A),list(A),rotate(A,modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% rotate_conv_mod
tff(fact_8050_div__gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( ( A2 != zero_zero(A) )
            | ( B2 != zero_zero(A) ) )
         => algebr8660921524188924756oprime(A,divide_divide(A,A2,gcd_gcd(A,A2,B2)),divide_divide(A,B2,gcd_gcd(A,A2,B2))) ) ) ).

% div_gcd_coprime
tff(fact_8051_rotate__def,axiom,
    ! [A: $tType,Nb: nat] : rotate(A,Nb) = aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(nat,fun(fun(list(A),list(A)),fun(list(A),list(A))),compow(fun(list(A),list(A))),Nb),rotate1(A)) ).

% rotate_def
tff(fact_8052_mult__mod__cancel__left,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [Nb: A,A2: A,Mb: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),Nb),A2),Mb) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),Nb),B2),Mb) )
         => ( algebr8660921524188924756oprime(A,Mb,Nb)
           => ( modulo_modulo(A,A2,Mb) = modulo_modulo(A,B2,Mb) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_8053_mult__mod__cancel__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [A2: A,Nb: A,Mb: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),Nb),Mb) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),Nb),Mb) )
         => ( algebr8660921524188924756oprime(A,Mb,Nb)
           => ( modulo_modulo(A,A2,Mb) = modulo_modulo(A,B2,Mb) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_8054_coprime__dvd__mult__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C3)
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> dvd_dvd(A,A2,B2) ) ) ) ).

% coprime_dvd_mult_right_iff
tff(fact_8055_coprime__dvd__mult__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C3)
         => ( dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
          <=> dvd_dvd(A,A2,B2) ) ) ) ).

% coprime_dvd_mult_left_iff
tff(fact_8056_divides__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C3: A,B2: A] :
          ( dvd_dvd(A,A2,C3)
         => ( dvd_dvd(A,B2,C3)
           => ( algebr8660921524188924756oprime(A,A2,B2)
             => dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) ) ) ) ) ).

% divides_mult
tff(fact_8057_coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ! [C2: A] :
              ( dvd_dvd(A,C2,A2)
             => ( dvd_dvd(A,C2,B2)
               => dvd_dvd(A,C2,one_one(A)) ) )
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprimeI
tff(fact_8058_coprime__def,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
        <=> ! [C4: A] :
              ( dvd_dvd(A,C4,A2)
             => ( dvd_dvd(A,C4,B2)
               => dvd_dvd(A,C4,one_one(A)) ) ) ) ) ).

% coprime_def
tff(fact_8059_not__coprimeE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ algebr8660921524188924756oprime(A,A2,B2)
         => ~ ! [C2: A] :
                ( dvd_dvd(A,C2,A2)
               => ( dvd_dvd(A,C2,B2)
                 => dvd_dvd(A,C2,one_one(A)) ) ) ) ) ).

% not_coprimeE
tff(fact_8060_not__coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,A2: A,B2: A] :
          ( dvd_dvd(A,C3,A2)
         => ( dvd_dvd(A,C3,B2)
           => ( ~ dvd_dvd(A,C3,one_one(A))
             => ~ algebr8660921524188924756oprime(A,A2,B2) ) ) ) ) ).

% not_coprimeI
tff(fact_8061_coprime__absorb__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xb: A,Y: A] :
          ( dvd_dvd(A,Xb,Y)
         => ( algebr8660921524188924756oprime(A,Xb,Y)
          <=> dvd_dvd(A,Xb,one_one(A)) ) ) ) ).

% coprime_absorb_left
tff(fact_8062_coprime__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,D3: A,A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C3,D3)
         => ( ! [E2: A] :
                ( ~ dvd_dvd(A,E2,one_one(A))
               => ( dvd_dvd(A,E2,A2)
                 => ( dvd_dvd(A,E2,B2)
                   => dvd_dvd(A,E2,C3) ) ) )
           => ( ! [E2: A] :
                  ( ~ dvd_dvd(A,E2,one_one(A))
                 => ( dvd_dvd(A,E2,A2)
                   => ( dvd_dvd(A,E2,B2)
                     => dvd_dvd(A,E2,D3) ) ) )
             => algebr8660921524188924756oprime(A,A2,B2) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_8063_coprime__absorb__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Y: A,Xb: A] :
          ( dvd_dvd(A,Y,Xb)
         => ( algebr8660921524188924756oprime(A,Xb,Y)
          <=> dvd_dvd(A,Y,one_one(A)) ) ) ) ).

% coprime_absorb_right
tff(fact_8064_coprime__common__divisor,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
         => ( dvd_dvd(A,C3,A2)
           => ( dvd_dvd(A,C3,B2)
             => dvd_dvd(A,C3,one_one(A)) ) ) ) ) ).

% coprime_common_divisor
tff(fact_8065_is__unit__left__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( dvd_dvd(A,A2,one_one(A))
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% is_unit_left_imp_coprime
tff(fact_8066_is__unit__right__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( dvd_dvd(A,B2,one_one(A))
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% is_unit_right_imp_coprime
tff(fact_8067_rotate1__rotate__swap,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate1(A),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),list(A),rotate(A,Nb),aa(list(A),list(A),rotate1(A),Xs)) ).

% rotate1_rotate_swap
tff(fact_8068_coprime__Suc__right__nat,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,Nb,aa(nat,nat,suc,Nb)) ).

% coprime_Suc_right_nat
tff(fact_8069_coprime__Suc__left__nat,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,Nb),Nb) ).

% coprime_Suc_left_nat
tff(fact_8070_coprime__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( algebr8660921524188924756oprime(A,B2,A2)
        <=> algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% coprime_commute
tff(fact_8071_coprime__crossproduct__nat,axiom,
    ! [A2: nat,D3: nat,B2: nat,C3: nat] :
      ( algebr8660921524188924756oprime(nat,A2,D3)
     => ( algebr8660921524188924756oprime(nat,B2,C3)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),C3) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),D3) )
        <=> ( ( A2 = B2 )
            & ( C3 = D3 ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_8072_coprime__1__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,one_one(A),A2) ) ).

% coprime_1_left
tff(fact_8073_coprime__1__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,A2,one_one(A)) ) ).

% coprime_1_right
tff(fact_8074_rotate__rotate,axiom,
    ! [A: $tType,Mb: nat,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Mb),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb)),Xs) ).

% rotate_rotate
tff(fact_8075_coprime__divisors,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A,B2: A,D3: A] :
          ( dvd_dvd(A,A2,C3)
         => ( dvd_dvd(A,B2,D3)
           => ( algebr8660921524188924756oprime(A,C3,D3)
             => algebr8660921524188924756oprime(A,A2,B2) ) ) ) ) ).

% coprime_divisors
tff(fact_8076_coprime__Suc__0__right,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,Nb,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_8077_coprime__Suc__0__left,axiom,
    ! [Nb: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) ).

% coprime_Suc_0_left
tff(fact_8078_coprime__add__one__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),A2) ) ).

% coprime_add_one_left
tff(fact_8079_coprime__add__one__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : algebr8660921524188924756oprime(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))) ) ).

% coprime_add_one_right
tff(fact_8080_coprime__common__divisor__nat,axiom,
    ! [A2: nat,B2: nat,Xb: nat] :
      ( algebr8660921524188924756oprime(nat,A2,B2)
     => ( dvd_dvd(nat,Xb,A2)
       => ( dvd_dvd(nat,Xb,B2)
         => ( Xb = one_one(nat) ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_8081_gcd__eq__1__imp__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( gcd_gcd(A,A2,B2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A2,B2) ) ) ).

% gcd_eq_1_imp_coprime
tff(fact_8082_coprime__iff__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,B2)
        <=> ( gcd_gcd(A,A2,B2) = one_one(A) ) ) ) ).

% coprime_iff_gcd_eq_1
tff(fact_8083_gcd__mult__right__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C3)
         => ( gcd_gcd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = gcd_gcd(A,A2,B2) ) ) ) ).

% gcd_mult_right_right_cancel
tff(fact_8084_gcd__mult__right__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,C3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A2,C3)
         => ( gcd_gcd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = gcd_gcd(A,A2,B2) ) ) ) ).

% gcd_mult_right_left_cancel
tff(fact_8085_gcd__mult__left__right__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C3: A,A2: A] :
          ( algebr8660921524188924756oprime(A,B2,C3)
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),B2) = gcd_gcd(A,A2,B2) ) ) ) ).

% gcd_mult_left_right_cancel
tff(fact_8086_gcd__mult__left__left__cancel,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C3: A,A2: A] :
          ( algebr8660921524188924756oprime(A,B2,C3)
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2),B2) = gcd_gcd(A,A2,B2) ) ) ) ).

% gcd_mult_left_left_cancel
tff(fact_8087_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( gcd_gcd(A,A2,B2) != zero_zero(A) )
         => ? [A9: A,B7: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),A9),gcd_gcd(A,A2,B2)) )
              & ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B7),gcd_gcd(A,A2,B2)) )
              & algebr8660921524188924756oprime(A,A9,B7) ) ) ) ).

% gcd_coprime_exists
tff(fact_8088_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,A3: A,B3: A] :
          ( ( gcd_gcd(A,A2,B2) != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),gcd_gcd(A,A2,B2)) )
           => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),gcd_gcd(A,A2,B2)) )
             => algebr8660921524188924756oprime(A,A3,B3) ) ) ) ) ).

% gcd_coprime
tff(fact_8089_invertible__coprime,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C3) = one_one(A) )
         => algebr8660921524188924756oprime(A,A2,C3) ) ) ).

% invertible_coprime
tff(fact_8090_rotate__map,axiom,
    ! [A: $tType,B: $tType,Nb: nat,F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate(A,Nb),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),rotate(B,Nb),Xs)) ).

% rotate_map
tff(fact_8091_coprime__diff__one__left__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => algebr8660921524188924756oprime(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat)),Nb) ) ).

% coprime_diff_one_left_nat
tff(fact_8092_coprime__diff__one__right__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Nb)
     => algebr8660921524188924756oprime(nat,Nb,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Nb),one_one(nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_8093_Rats__abs__nat__div__natE,axiom,
    ! [Xb: real] :
      ( member(real,Xb,field_char_0_Rats(real))
     => ~ ! [M: nat,N: nat] :
            ( ( N != zero_zero(nat) )
           => ( ( aa(real,real,abs_abs(real),Xb) = divide_divide(real,aa(nat,real,semiring_1_of_nat(real),M),aa(nat,real,semiring_1_of_nat(real),N)) )
             => ~ algebr8660921524188924756oprime(nat,M,N) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_8094_rotate__rev,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,Nb),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs)))),Xs)) ).

% rotate_rev
tff(fact_8095_nth__rotate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Mb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,Mb),Xs)),Nb) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Mb),Nb),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_8096_hd__rotate__conv__nth,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),rotate(A,Nb),Xs)) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,Nb,aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% hd_rotate_conv_nth
tff(fact_8097_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(C,A),G3: fun(C,B),A5: set(C)] : image(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_ail(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F3),G3),A5) = product_Sigma(A,B,image(C,A,F3,A5),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_aim(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F3),G3),A5)) ).

% image_split_eq_Sigma
tff(fact_8098_ntrancl__Suc,axiom,
    ! [A: $tType,Nb: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,aa(nat,nat,suc,Nb),R) = relcomp(A,A,A,transitive_ntrancl(A,Nb,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))),id2(A)),R)) ).

% ntrancl_Suc
tff(fact_8099_coprime__abs__left__iff,axiom,
    ! [K2: int,L: int] :
      ( algebr8660921524188924756oprime(int,aa(int,int,abs_abs(int),K2),L)
    <=> algebr8660921524188924756oprime(int,K2,L) ) ).

% coprime_abs_left_iff
tff(fact_8100_coprime__abs__right__iff,axiom,
    ! [K2: int,L: int] :
      ( algebr8660921524188924756oprime(int,K2,aa(int,int,abs_abs(int),L))
    <=> algebr8660921524188924756oprime(int,K2,L) ) ).

% coprime_abs_right_iff
tff(fact_8101_pair__in__Id__conv,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( 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),id2(A))
    <=> ( A2 = B2 ) ) ).

% pair_in_Id_conv
tff(fact_8102_IdI,axiom,
    ! [A: $tType,A2: A] : 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),id2(A)) ).

% IdI
tff(fact_8103_coprime__int__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),Mb),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> algebr8660921524188924756oprime(nat,Mb,Nb) ) ).

% coprime_int_iff
tff(fact_8104_coprime__nat__abs__left__iff,axiom,
    ! [K2: int,Nb: nat] :
      ( algebr8660921524188924756oprime(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)),Nb)
    <=> algebr8660921524188924756oprime(int,K2,aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% coprime_nat_abs_left_iff
tff(fact_8105_coprime__nat__abs__right__iff,axiom,
    ! [Nb: nat,K2: int] :
      ( algebr8660921524188924756oprime(nat,Nb,aa(int,nat,nat2,aa(int,int,abs_abs(int),K2)))
    <=> algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),Nb),K2) ) ).

% coprime_nat_abs_right_iff
tff(fact_8106_vimage__fst,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : vimage(product_prod(A,B),A,product_fst(A,B),A5) = product_Sigma(A,B,A5,aTP_Lamp_ahh(A,set(B))) ).

% vimage_fst
tff(fact_8107_vimage__snd,axiom,
    ! [A: $tType,B: $tType,A5: set(B)] : vimage(product_prod(A,B),B,product_snd(A,B),A5) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ahf(set(B),fun(A,set(B)),A5)) ).

% vimage_snd
tff(fact_8108_vimage__Times,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,product_prod(B,C)),A5: set(B),B5: set(C)] : vimage(A,product_prod(B,C),F3,product_Sigma(B,C,A5,aTP_Lamp_ain(set(C),fun(B,set(C)),B5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),F3),A5)),vimage(A,C,aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),F3),B5)) ).

% vimage_Times
tff(fact_8109_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,Xb: B,A5: set(B),F3: fun(B,set(A))] :
      vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),product_Sigma(B,A,A5,F3)) = $ite(member(B,Xb,A5),aa(B,set(A),F3,Xb),bot_bot(set(A))) ).

% Pair_vimage_Sigma
tff(fact_8110_coprime__common__divisor__int,axiom,
    ! [A2: int,B2: int,Xb: int] :
      ( algebr8660921524188924756oprime(int,A2,B2)
     => ( dvd_dvd(int,Xb,A2)
       => ( dvd_dvd(int,Xb,B2)
         => ( aa(int,int,abs_abs(int),Xb) = one_one(int) ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_8111_coprime__crossproduct__int,axiom,
    ! [A2: int,D3: int,B2: int,C3: int] :
      ( algebr8660921524188924756oprime(int,A2,D3)
     => ( algebr8660921524188924756oprime(int,B2,C3)
       => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A2)),aa(int,int,abs_abs(int),C3)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),B2)),aa(int,int,abs_abs(int),D3)) )
        <=> ( ( aa(int,int,abs_abs(int),A2) = aa(int,int,abs_abs(int),B2) )
            & ( aa(int,int,abs_abs(int),C3) = aa(int,int,abs_abs(int),D3) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_8112_vimage__Suc__insert__Suc,axiom,
    ! [Nb: nat,A5: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,Nb)),A5)) = aa(set(nat),set(nat),insert(nat,Nb),vimage(nat,nat,suc,A5)) ).

% vimage_Suc_insert_Suc
tff(fact_8113_vimage__Suc__insert__0,axiom,
    ! [A5: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),insert(nat,zero_zero(nat)),A5)) = vimage(nat,nat,suc,A5) ).

% vimage_Suc_insert_0
tff(fact_8114_IdD,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( 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),id2(A))
     => ( A2 = B2 ) ) ).

% IdD
tff(fact_8115_IdE,axiom,
    ! [A: $tType,P2: product_prod(A,A)] :
      ( member(product_prod(A,A),P2,id2(A))
     => ~ ! [X4: A] : P2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).

% IdE
tff(fact_8116_finite__vimage__Suc__iff,axiom,
    ! [F4: set(nat)] :
      ( aa(set(nat),$o,finite_finite(nat),vimage(nat,nat,suc,F4))
    <=> aa(set(nat),$o,finite_finite(nat),F4) ) ).

% finite_vimage_Suc_iff
tff(fact_8117_Id__def,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aio(product_prod(A,A),$o)) ).

% Id_def
tff(fact_8118_irrefl__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : irrefl(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))),minus_minus(set(product_prod(A,A))),R2),id2(A))) ).

% irrefl_diff_Id
tff(fact_8119_vimage__Diff,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(B),B5: set(B)] : vimage(A,B,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),vimage(A,B,F3,A5)),vimage(A,B,F3,B5)) ).

% vimage_Diff
tff(fact_8120_rtrancl__r__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : 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))),minus_minus(set(product_prod(A,A))),R2),id2(A))) = transitive_rtrancl(A,R2) ).

% rtrancl_r_diff_Id
tff(fact_8121_reflcl__set__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X: A,Xa: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),aTP_Lamp_abu(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),fequal(A)),X),Xa)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa),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))),R2),id2(A))) ) ).

% reflcl_set_eq
tff(fact_8122_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),D4: set(A),A5: set(B)] :
      ( inj_on(A,B,F3,D4)
     => ( aa(set(B),$o,finite_finite(B),A5)
       => 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)),inf_inf(set(A)),vimage(A,B,F3,A5)),D4))),aa(set(B),nat,finite_card(B),A5)) ) ) ).

% card_vimage_inj_on_le
tff(fact_8123_Rats__cases_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xb: A] :
          ( member(A,Xb,field_char_0_Rats(A))
         => ~ ! [A4: int,B4: int] :
                ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B4)
               => ( algebr8660921524188924756oprime(int,A4,B4)
                 => ( Xb != divide_divide(A,aa(int,A,ring_1_of_int(A),A4),aa(int,A,ring_1_of_int(A),B4)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_8124_set__decode__div__2,axiom,
    ! [Xb: nat] : nat_set_decode(divide_divide(nat,Xb,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = vimage(nat,nat,suc,nat_set_decode(Xb)) ).

% set_decode_div_2
tff(fact_8125_set__encode__vimage__Suc,axiom,
    ! [A5: set(nat)] : aa(set(nat),nat,nat_set_encode,vimage(nat,nat,suc,A5)) = divide_divide(nat,aa(set(nat),nat,nat_set_encode,A5),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% set_encode_vimage_Suc
tff(fact_8126_inv__image__partition,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ! [X4: A] :
          ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X4) )
     => ( ! [Y3: A] :
            ( member(A,Y3,aa(list(A),set(A),set2(A),Ys))
           => ~ aa(A,$o,P,Y3) )
       => ( vimage(list(A),product_prod(list(A),list(A)),partition(A,P),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),insert(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)),bot_bot(set(product_prod(list(A),list(A)))))) = shuffles(A,Xs,Ys) ) ) ) ).

% inv_image_partition
tff(fact_8127_listrel1__subset__listrel,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),R3)
     => ( refl_on(A,top_top(set(A)),R3)
       => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel(A,A,R3)) ) ) ).

% listrel1_subset_listrel
tff(fact_8128_total__on__singleton,axiom,
    ! [A: $tType,Xb: A] : total_on(A,aa(set(A),set(A),insert(A,Xb),bot_bot(set(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),Xb),Xb)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_8129_total__on__diff__Id,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A5,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))),minus_minus(set(product_prod(A,A))),R2),id2(A)))
    <=> total_on(A,A5,R2) ) ).

% total_on_diff_Id
tff(fact_8130_total__lenlex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,top_top(set(A)),R2)
     => total_on(list(A),top_top(set(list(A))),lenlex(A,R2)) ) ).

% total_lenlex
tff(fact_8131_total__lexord,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,top_top(set(A)),R2)
     => total_on(list(A),top_top(set(list(A))),lexord(A,R2)) ) ).

% total_lexord
tff(fact_8132_total__onI,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( ! [X4: A,Y3: A] :
          ( member(A,X4,A5)
         => ( member(A,Y3,A5)
           => ( ( X4 != Y3 )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R2)
                | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),R2) ) ) ) )
     => total_on(A,A5,R2) ) ).

% total_onI
tff(fact_8133_total__on__def,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A5,R2)
    <=> ! [X3: A] :
          ( member(A,X3,A5)
         => ! [Xa2: A] :
              ( member(A,Xa2,A5)
             => ( ( X3 != Xa2 )
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa2),R2)
                  | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X3),R2) ) ) ) ) ) ).

% total_on_def
tff(fact_8134_refl__onD,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),A2: A] :
      ( refl_on(A,A5,R2)
     => ( member(A,A2,A5)
       => 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),R2) ) ) ).

% refl_onD
tff(fact_8135_refl__onD1,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),Xb: A,Y: A] :
      ( refl_on(A,A5,R2)
     => ( 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)
       => member(A,Xb,A5) ) ) ).

% refl_onD1
tff(fact_8136_refl__onD2,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),Xb: A,Y: A] :
      ( refl_on(A,A5,R2)
     => ( 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)
       => member(A,Y,A5) ) ) ).

% refl_onD2
tff(fact_8137_refl__onI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_ahu(set(A),fun(A,set(A)),A5)))
     => ( ! [X4: A] :
            ( member(A,X4,A5)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R2) )
       => refl_on(A,A5,R2) ) ) ).

% refl_onI
tff(fact_8138_refl__on__def,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A5,R2)
    <=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_ahu(set(A),fun(A,set(A)),A5)))
        & ! [X3: A] :
            ( member(A,X3,A5)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2) ) ) ) ).

% refl_on_def
tff(fact_8139_refl__on__def_H,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A5,R2)
    <=> ( ! [X3: product_prod(A,A)] :
            ( member(product_prod(A,A),X3,R2)
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_aip(set(A),fun(A,fun(A,$o)),A5)),X3) )
        & ! [X3: A] :
            ( member(A,X3,A5)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R2) ) ) ) ).

% refl_on_def'
tff(fact_8140_refl__on__singleton,axiom,
    ! [A: $tType,Xb: A] : refl_on(A,aa(set(A),set(A),insert(A,Xb),bot_bot(set(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),Xb),Xb)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_8141_refl__on__domain,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( refl_on(A,A5,R2)
     => ( 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)
       => ( member(A,A2,A5)
          & member(A,B2,A5) ) ) ) ).

% refl_on_domain
tff(fact_8142_prod_Oset__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A),Xs: list(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),aa(list(B),set(B),set2(B),Xs)) = groups5270119922927024881d_list(A,aa(list(B),list(A),map(B,A,G3),remdups(B,Xs))) ) ).

% prod.set_conv_list
tff(fact_8143_prod__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A,Xs: list(A)] : groups5270119922927024881d_list(A,aa(list(A),list(A),cons(A,Xb),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),groups5270119922927024881d_list(A,Xs)) ) ).

% prod_list.Cons
tff(fact_8144_prod__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( groups5270119922927024881d_list(A,nil(A)) = one_one(A) ) ) ).

% prod_list.Nil
tff(fact_8145_prod__list_Oappend,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A),Ys: list(A)] : groups5270119922927024881d_list(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),times_times(A),groups5270119922927024881d_list(A,Xs)),groups5270119922927024881d_list(A,Ys)) ) ).

% prod_list.append
tff(fact_8146_prod__list__coprime__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),A2: A] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => algebr8660921524188924756oprime(A,X4,A2) )
         => algebr8660921524188924756oprime(A,groups5270119922927024881d_list(A,Xs),A2) ) ) ).

% prod_list_coprime_left
tff(fact_8147_prod__list__coprime__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),A2: A] :
          ( ! [X4: A] :
              ( member(A,X4,aa(list(A),set(A),set2(A),Xs))
             => algebr8660921524188924756oprime(A,A2,X4) )
         => algebr8660921524188924756oprime(A,A2,groups5270119922927024881d_list(A,Xs)) ) ) ).

% prod_list_coprime_right
tff(fact_8148_prod__list__zero__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Xs: list(A)] :
          ( ( groups5270119922927024881d_list(A,Xs) = zero_zero(A) )
        <=> member(A,zero_zero(A),aa(list(A),set(A),set2(A),Xs)) ) ) ).

% prod_list_zero_iff
tff(fact_8149_prod__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A)] : groups5270119922927024881d_list(A,Xs) = aa(A,A,foldr(A,A,times_times(A),Xs),one_one(A)) ) ).

% prod_list.eq_foldr
tff(fact_8150_prod_Odistinct__set__conv__list,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Xs: list(A),G3: fun(A,B)] :
          ( distinct(A,Xs)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(list(A),set(A),set2(A),Xs)) = groups5270119922927024881d_list(B,aa(list(A),list(B),map(A,B,G3),Xs)) ) ) ) ).

% prod.distinct_set_conv_list
tff(fact_8151_linear__order__on__singleton,axiom,
    ! [A: $tType,Xb: A] : order_679001287576687338der_on(A,aa(set(A),set(A),insert(A,Xb),bot_bot(set(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),Xb),Xb)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_8152_Total__subset__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R2),R2)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),id2(A))
       => ( ( R2 = bot_bot(set(product_prod(A,A))) )
          | ? [A4: A] : R2 = 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),A4),A4)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_8153_Field__insert,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : field2(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)),R2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),field2(A,R2)) ).

% Field_insert
tff(fact_8154_Linear__order__in__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( member(A,A2,field2(A,R2))
       => ( member(A,B2,field2(A,R2))
         => ( 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)
          <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A2),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))),minus_minus(set(product_prod(A,A))),R2),id2(A))) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_8155_FieldI1,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R)
     => member(A,I,field2(A,R)) ) ).

% FieldI1
tff(fact_8156_FieldI2,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R)
     => member(A,J,field2(A,R)) ) ).

% FieldI2
tff(fact_8157_Total__Id__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R2),R2)
     => ( ~ aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),id2(A))
       => ( field2(A,R2) = field2(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))),minus_minus(set(product_prod(A,A))),R2),id2(A))) ) ) ) ).

% Total_Id_Field
tff(fact_8158_UnderS__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : order_UnderS(A,R2,A5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aiq(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A5)) ).

% UnderS_def
tff(fact_8159_Under__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : order_Under(A,R2,A5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_air(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A5)) ).

% Under_def
tff(fact_8160_strict__linear__order__on__diff__Id,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,A5,R2)
     => order_5396836661320670305der_on(A,A5,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))),minus_minus(set(product_prod(A,A))),R2),id2(A))) ) ).

% strict_linear_order_on_diff_Id
tff(fact_8161_Above__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A)] : order_Above(A,R2,A5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ais(set(product_prod(A,A)),fun(set(A),fun(A,$o)),R2),A5)) ).

% Above_def
tff(fact_8162_Field__natLeq__on,axiom,
    ! [Nb: nat] : field2(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_ait(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Nb)) ).

% Field_natLeq_on
tff(fact_8163_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R2),R2)
     => ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R2),id2(A)))
      <=> ! [A10: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A10),field2(A,R2))
           => ( ( A10 != bot_bot(set(A)) )
             => ? [X3: A] :
                  ( member(A,X3,A10)
                  & ! [Xa2: A] :
                      ( member(A,Xa2,A10)
                     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa2),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_8164_wf__listrel1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(list(A),listrel1(A,R2))
    <=> wf(A,R2) ) ).

% wf_listrel1_iff
tff(fact_8165_wf__lex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lex(A,R2)) ) ).

% wf_lex
tff(fact_8166_wf__lenlex,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
     => wf(list(A),lenlex(A,R2)) ) ).

% wf_lenlex
tff(fact_8167_wf__insert,axiom,
    ! [A: $tType,Y: A,Xb: A,R2: set(product_prod(A,A))] :
      ( wf(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)),R2))
    <=> ( wf(A,R2)
        & ~ 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)) ) ) ).

% wf_insert
tff(fact_8168_wfI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A5: set(A),B5: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A5,aTP_Lamp_ahu(set(A),fun(A,set(A)),B5)))
     => ( ! [X4: A,P5: fun(A,$o)] :
            ( ! [Xa: A] :
                ( ! [Y3: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa),R2)
                   => aa(A,$o,P5,Y3) )
               => aa(A,$o,P5,Xa) )
           => ( member(A,X4,A5)
             => ( member(A,X4,B5)
               => aa(A,$o,P5,X4) ) ) )
       => wf(A,R2) ) ) ).

% wfI
tff(fact_8169_wf__less,axiom,
    wf(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),ord_less(nat)))) ).

% wf_less
tff(fact_8170_wf,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wf(A,aa(fun(product_prod(A,A),$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),ord_less(A)))) ) ).

% wf
tff(fact_8171_wf__if__measure,axiom,
    ! [A: $tType,P: fun(A,$o),F3: fun(A,nat),G3: fun(A,A)] :
      ( ! [X4: A] :
          ( aa(A,$o,P,X4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,aa(A,A,G3,X4))),aa(A,nat,F3,X4)) )
     => wf(A,aa(fun(product_prod(A,A),$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,$o),aa(fun(A,A),fun(A,fun(A,$o)),aTP_Lamp_aiu(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),P),G3)))) ) ).

% wf_if_measure
tff(fact_8172_wf__bounded__measure,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,nat),F3: fun(A,nat)] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4),R2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Ub,B4)),aa(A,nat,Ub,A4))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,B4)),aa(A,nat,Ub,A4))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,A4)),aa(A,nat,F3,B4)) ) )
     => wf(A,R2) ) ).

% wf_bounded_measure
tff(fact_8173_wf__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [P6: fun(A,$o)] :
          ( ! [X3: A] :
              ( ! [Y5: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R2)
                 => aa(A,$o,P6,Y5) )
             => aa(A,$o,P6,X3) )
         => ! [X_12: A] : aa(A,$o,P6,X_12) ) ) ).

% wf_def
tff(fact_8174_wfE__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xb: A,Q: set(A)] :
      ( wf(A,R)
     => ( member(A,Xb,Q)
       => ~ ! [Z3: A] :
              ( member(A,Z3,Q)
             => ~ ! [Y4: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R)
                   => ~ member(A,Y4,Q) ) ) ) ) ).

% wfE_min
tff(fact_8175_wfI__min,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X4: A,Q7: set(A)] :
          ( member(A,X4,Q7)
         => ? [Xa: A] :
              ( member(A,Xa,Q7)
              & ! [Y3: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa),R)
                 => ~ member(A,Y3,Q7) ) ) )
     => wf(A,R) ) ).

% wfI_min
tff(fact_8176_wfUNIVI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [P5: fun(A,$o),X4: A] :
          ( ! [Xa: A] :
              ( ! [Y3: A] :
                  ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Xa),R2)
                 => aa(A,$o,P5,Y3) )
             => aa(A,$o,P5,Xa) )
         => aa(A,$o,P5,X4) )
     => wf(A,R2) ) ).

% wfUNIVI
tff(fact_8177_wf__asym,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A,Xb: A] :
      ( wf(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Xb),R2)
       => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),A2),R2) ) ) ).

% wf_asym
tff(fact_8178_wf__induct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A2: A] :
      ( wf(A,R2)
     => ( ! [X4: A] :
            ( ! [Y4: A] :
                ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X4),R2)
               => aa(A,$o,P,Y4) )
           => aa(A,$o,P,X4) )
       => aa(A,$o,P,A2) ) ) ).

% wf_induct
tff(fact_8179_wf__irrefl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A2: A] :
      ( wf(A,R2)
     => ~ 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),R2) ) ).

% wf_irrefl
tff(fact_8180_ATP_Olambda__1,axiom,
    ! [Uu2: product_prod(int,int)] :
      aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_ov(product_prod(int,int),product_prod(int,int)),Uu2) = $ite(aa(product_prod(int,int),int,product_fst(int,int),Uu2) = 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(product_prod(int,int),int,product_snd(int,int),Uu2)),aa(product_prod(int,int),int,product_fst(int,int),Uu2))) ).

% ATP.lambda_1
tff(fact_8181_ATP_Olambda__2,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_ea(nat,real),Uu2) = 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))),Uu2)),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),Uu2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat)))) ).

% ATP.lambda_2
tff(fact_8182_ATP_Olambda__3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_wc(A,A),Uu2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Uu2)),one_one(A)),Uu2) ) ).

% ATP.lambda_3
tff(fact_8183_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu2: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_ps(set(set(A)),int),Uu2) = 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)),Uu2)),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)),Uu2)))) ).

% ATP.lambda_4
tff(fact_8184_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu2: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_adu(A,set(product_prod(A,A))),Uu2) = 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),Uu2),Uu2)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_5
tff(fact_8185_ATP_Olambda__6,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_ek(nat,real),Uu2) = aa(nat,real,aa(real,fun(nat,real),power_power(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,nat,suc,Uu2)) ).

% ATP.lambda_6
tff(fact_8186_ATP_Olambda__7,axiom,
    ! [Uu2: real] :
      ( aa(real,$o,aTP_Lamp_iy(real,$o),Uu2)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu2)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))
        & ( cos(real,Uu2) = zero_zero(real) ) ) ) ).

% ATP.lambda_7
tff(fact_8187_ATP_Olambda__8,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: product_prod(int,int)] : aa(product_prod(int,int),A,aTP_Lamp_os(product_prod(int,int),A),Uu2) = divide_divide(A,aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_fst(int,int),Uu2)),aa(int,A,ring_1_of_int(A),aa(product_prod(int,int),int,product_snd(int,int),Uu2))) ) ).

% ATP.lambda_8
tff(fact_8188_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_xg(nat,A),Uu2) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu2)),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_9
tff(fact_8189_ATP_Olambda__10,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_dz(nat,real),Uu2) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu2)) ).

% ATP.lambda_10
tff(fact_8190_ATP_Olambda__11,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_xf(nat,A),Uu2) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uu2),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu2))) ) ).

% ATP.lambda_11
tff(fact_8191_ATP_Olambda__12,axiom,
    ! [Uu2: real] : aa(real,real,aTP_Lamp_wf(real,real),Uu2) = divide_divide(real,cos(real,Uu2),sin(real,Uu2)) ).

% ATP.lambda_12
tff(fact_8192_ATP_Olambda__13,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_ws(nat,real),Uu2) = aa(real,real,root(Uu2),aa(nat,real,semiring_1_of_nat(real),Uu2)) ).

% ATP.lambda_13
tff(fact_8193_ATP_Olambda__14,axiom,
    ! [Uu2: nat] : aa(nat,nat,aTP_Lamp_ma(nat,nat),Uu2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_14
tff(fact_8194_ATP_Olambda__15,axiom,
    ! [B: $tType,Uu2: B] : aa(B,product_prod(B,B),aTP_Lamp_ly(B,product_prod(B,B)),Uu2) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu2),Uu2) ).

% ATP.lambda_15
tff(fact_8195_ATP_Olambda__16,axiom,
    ! [A: $tType,Uu2: A] : aa(A,product_prod(A,A),aTP_Lamp_lx(A,product_prod(A,A)),Uu2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),Uu2) ).

% ATP.lambda_16
tff(fact_8196_ATP_Olambda__17,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_ez(A,A),Uu2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),one_one(A)) ) ).

% ATP.lambda_17
tff(fact_8197_ATP_Olambda__18,axiom,
    ! [A: $tType,Uu2: A] : aa(A,list(A),aTP_Lamp_pm(A,list(A)),Uu2) = aa(list(A),list(A),cons(A,Uu2),nil(A)) ).

% ATP.lambda_18
tff(fact_8198_ATP_Olambda__19,axiom,
    ! [Uu2: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_oq(product_prod(int,int),$o),Uu2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu2)),aa(product_prod(int,int),int,product_snd(int,int),Uu2))) ) ).

% ATP.lambda_19
tff(fact_8199_ATP_Olambda__20,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_wz(nat,real),Uu2) = divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),Uu2)) ).

% ATP.lambda_20
tff(fact_8200_ATP_Olambda__21,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_xe(nat,A),Uu2) = divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_21
tff(fact_8201_ATP_Olambda__22,axiom,
    ! [B: $tType,Uu2: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_afc(list(B),fun(nat,nat)),Uu2) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Uu2)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_22
tff(fact_8202_ATP_Olambda__23,axiom,
    ! [A: $tType,Uu2: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_adq(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu2) = insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),Uu2)) ).

% ATP.lambda_23
tff(fact_8203_ATP_Olambda__24,axiom,
    ! [B: $tType,Uu2: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_afd(list(B),$o),Uu2)
    <=> ( Uu2 != nil(B) ) ) ).

% ATP.lambda_24
tff(fact_8204_ATP_Olambda__25,axiom,
    ! [A: $tType,Uu2: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_aet(list(A),$o),Uu2)
    <=> ( Uu2 != nil(A) ) ) ).

% ATP.lambda_25
tff(fact_8205_ATP_Olambda__26,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,option(B))] : aa(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))),aTP_Lamp_afx(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Uu2) = aa(fun(A,fun(B,fun(A,option(B)))),fun(product_prod(A,B),fun(A,option(B))),product_case_prod(A,B,fun(A,option(B))),aTP_Lamp_afw(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu2)) ).

% ATP.lambda_26
tff(fact_8206_ATP_Olambda__27,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_adn(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu2) = aa(fun(A,fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(A,C),product_prod(A,product_prod(B,C))),product_case_prod(A,C,product_prod(A,product_prod(B,C))),aTP_Lamp_adm(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu2)) ).

% ATP.lambda_27
tff(fact_8207_ATP_Olambda__28,axiom,
    ! [Uu2: real] : aa(real,real,aTP_Lamp_sk(real,real),Uu2) = suminf(real,aTP_Lamp_co(real,fun(nat,real),Uu2)) ).

% ATP.lambda_28
tff(fact_8208_ATP_Olambda__29,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_xb(nat,real),Uu2) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu2))) ).

% ATP.lambda_29
tff(fact_8209_ATP_Olambda__30,axiom,
    ! [B: $tType,Uu2: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_afb(list(B),fun(nat,nat)),Uu2) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu2)) ).

% ATP.lambda_30
tff(fact_8210_ATP_Olambda__31,axiom,
    ! [A: $tType,Uu2: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_aeu(list(A),fun(nat,nat)),Uu2) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu2)) ).

% ATP.lambda_31
tff(fact_8211_ATP_Olambda__32,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: A] : aa(A,B,aTP_Lamp_abg(A,B),Uu2) = aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,Uu2)) ) ).

% ATP.lambda_32
tff(fact_8212_ATP_Olambda__33,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A)
        & ring_1(B)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: A] : aa(A,B,aTP_Lamp_abh(A,B),Uu2) = aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,Uu2)) ) ).

% ATP.lambda_33
tff(fact_8213_ATP_Olambda__34,axiom,
    ! [Uu2: num] : aa(num,option(num),aTP_Lamp_nd(num,option(num)),Uu2) = aa(num,option(num),some(num),aa(num,num,bit1,Uu2)) ).

% ATP.lambda_34
tff(fact_8214_ATP_Olambda__35,axiom,
    ! [Uu2: num] : aa(num,option(num),aTP_Lamp_mz(num,option(num)),Uu2) = aa(num,option(num),some(num),aa(num,num,bit0,Uu2)) ).

% ATP.lambda_35
tff(fact_8215_ATP_Olambda__36,axiom,
    ! [Uu2: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_cf(nat,fun(nat,product_prod(nat,nat))),Uu2) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu2)) ).

% ATP.lambda_36
tff(fact_8216_ATP_Olambda__37,axiom,
    ! [A: $tType,Uu2: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aTP_Lamp_aio(product_prod(A,A),$o),Uu2)
    <=> ? [X3: A] : Uu2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).

% ATP.lambda_37
tff(fact_8217_ATP_Olambda__38,axiom,
    ! [Uu2: real] :
      ( aa(real,$o,aTP_Lamp_nx(real,$o),Uu2)
    <=> ? [I3: int,N4: nat] :
          ( ( Uu2 = divide_divide(real,aa(int,real,ring_1_of_int(real),I3),aa(nat,real,semiring_1_of_nat(real),N4)) )
          & ( N4 != zero_zero(nat) ) ) ) ).

% ATP.lambda_38
tff(fact_8218_ATP_Olambda__39,axiom,
    ! [Uu2: real] :
      ( aa(real,$o,aTP_Lamp_ny(real,$o),Uu2)
    <=> ? [I3: int,J3: int] :
          ( ( Uu2 = divide_divide(real,aa(int,real,ring_1_of_int(real),I3),aa(int,real,ring_1_of_int(real),J3)) )
          & ( J3 != zero_zero(int) ) ) ) ).

% ATP.lambda_39
tff(fact_8219_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_zv(product_prod(A,A),$o),Uu2)
        <=> ? [X3: A,Y5: A] :
              ( ( Uu2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y5) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y5) ) ) ) ).

% ATP.lambda_40
tff(fact_8220_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_zu(product_prod(A,A),$o),Uu2)
        <=> ? [X3: A,Y5: A] :
              ( ( Uu2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y5) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X3) ) ) ) ).

% ATP.lambda_41
tff(fact_8221_ATP_Olambda__42,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_zs(product_prod(A,A),$o),Uu2)
        <=> ? [X3: A,Y5: A] :
              ( ( Uu2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y5) )
              & ( X3 != Y5 ) ) ) ) ).

% ATP.lambda_42
tff(fact_8222_ATP_Olambda__43,axiom,
    ! [Uu2: nat] : aa(nat,option(num),aTP_Lamp_nb(nat,option(num)),Uu2) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_43
tff(fact_8223_ATP_Olambda__44,axiom,
    ! [Uu2: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_ng(num,fun(nat,option(num)),Uu2),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_ne(nat,fun(num,option(num)),Uua),aTP_Lamp_nf(nat,fun(num,option(num)),Uua),Uu2) ).

% ATP.lambda_44
tff(fact_8224_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_iw(A,fun(nat,A),Uu2),Uua) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),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),Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_45
tff(fact_8225_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_hm(nat,fun(nat,A),Uu2),Uua) = $ite(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(Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_46
tff(fact_8226_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_iv(A,fun(nat,A),Uu2),Uua) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),zero_zero(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),Uu2),Uua))) ) ).

% ATP.lambda_47
tff(fact_8227_ATP_Olambda__48,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_el(fun(nat,real),fun(nat,real),Uu2),Uua) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),zero_zero(real),aa(nat,real,Uu2,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_48
tff(fact_8228_ATP_Olambda__49,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: product_prod(A,C),Uua: product_prod(C,B)] :
      aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_aec(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu2),Uua) = $ite(aa(product_prod(A,C),C,product_snd(A,C),Uu2) = aa(product_prod(C,B),C,product_fst(C,B),Uua),aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(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,C),A,product_fst(A,C),Uu2)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).

% ATP.lambda_49
tff(fact_8229_ATP_Olambda__50,axiom,
    ! [Uu2: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jj(int,fun(int,product_prod(int,int))),Uu2),Uua) = $ite(Uu2 = 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),Uu2)),Uua)),aa(int,int,abs_abs(int),Uu2))) ).

% ATP.lambda_50
tff(fact_8230_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_hn(nat,fun(nat,A),Uu2),Uua) = $ite(~ 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(Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_51
tff(fact_8231_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu2: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_acc(fun(A,A),fun(nat,A),Uu2),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),Uu2),top_top(A)) ) ).

% ATP.lambda_52
tff(fact_8232_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu2: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_aca(fun(A,A),fun(nat,A),Uu2),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),Uu2),bot_bot(A)) ) ).

% ATP.lambda_53
tff(fact_8233_ATP_Olambda__54,axiom,
    ! [Uu2: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_nh(nat,fun(num,option(num))),Uu2),Uua) = case_nat(option(num),none(num),aTP_Lamp_ng(num,fun(nat,option(num)),Uua),Uu2) ).

% ATP.lambda_54
tff(fact_8234_ATP_Olambda__55,axiom,
    ! [Uu2: nat,Uua: num] : aa(num,option(num),aTP_Lamp_ne(nat,fun(num,option(num)),Uu2),Uua) = case_option(option(num),num,none(num),aTP_Lamp_mz(num,option(num)),bit_take_bit_num(Uu2,Uua)) ).

% ATP.lambda_55
tff(fact_8235_ATP_Olambda__56,axiom,
    ! [Uu2: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_na(num,fun(nat,option(num)),Uu2),Uua) = case_option(option(num),num,none(num),aTP_Lamp_mz(num,option(num)),bit_take_bit_num(Uua,Uu2)) ).

% ATP.lambda_56
tff(fact_8236_ATP_Olambda__57,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_ln(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu2),Uua)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu2,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_57
tff(fact_8237_ATP_Olambda__58,axiom,
    ! [A: $tType,Uu2: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_aes(list(list(A)),fun(nat,list(A)),Uu2),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aer(list(list(A)),fun(nat,fun(nat,A)),Uu2),Uua)),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu2))) ).

% ATP.lambda_58
tff(fact_8238_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hu(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ht(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_59
tff(fact_8239_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gk(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_60
tff(fact_8240_ATP_Olambda__61,axiom,
    ! [Uu2: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_ou(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu2),Uua) = 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(product_prod(int,int),int,product_fst(int,int),Uu2)),aa(product_prod(int,int),int,product_snd(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uu2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu2)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_61
tff(fact_8241_ATP_Olambda__62,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_cn(real,fun(nat,real),Uu2),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)),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),Uu2),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_62
tff(fact_8242_ATP_Olambda__63,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_et(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(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),Uu2),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_63
tff(fact_8243_ATP_Olambda__64,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ew(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(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),Uu2),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_64
tff(fact_8244_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hz(nat,fun(nat,A),Uu2),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(Uu2),Uua))) ) ).

% ATP.lambda_65
tff(fact_8245_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hg(nat,fun(nat,A),Uu2),Uua) = 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),Uu2),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_66
tff(fact_8246_ATP_Olambda__67,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_iz(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
        & ( sin(real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_67
tff(fact_8247_ATP_Olambda__68,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ix(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(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),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
        & ( aa(real,real,tan(real),Uua) = Uu2 ) ) ) ).

% ATP.lambda_68
tff(fact_8248_ATP_Olambda__69,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_kq(code_integer,fun(code_integer,int)),Uu2),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(Uu2)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_69
tff(fact_8249_ATP_Olambda__70,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_cj(nat,fun(nat,a)),Uu2),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),Uu2)),
        $ite(Uua = zero_zero(nat),m3,aa(a,a,aa(a,fun(a,a),plus_plus(a),m3),one_one(a))) ) ).

% ATP.lambda_70
tff(fact_8250_ATP_Olambda__71,axiom,
    ! [Uu2: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_ow(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu2),Uua) = 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(product_prod(int,int),int,product_fst(int,int),Uu2)),aa(product_prod(int,int),int,product_fst(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu2)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).

% ATP.lambda_71
tff(fact_8251_ATP_Olambda__72,axiom,
    ! [Uu2: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_je(real,fun(int,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu2)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu2),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_72
tff(fact_8252_ATP_Olambda__73,axiom,
    ! [Uu2: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_jf(rat,fun(int,$o),Uu2),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu2)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu2),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_73
tff(fact_8253_ATP_Olambda__74,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_co(real,fun(nat,real),Uu2),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),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),Uu2),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_74
tff(fact_8254_ATP_Olambda__75,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_sl(real,fun(nat,real),Uu2),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),Uu2),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_75
tff(fact_8255_ATP_Olambda__76,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),Uu2),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,Uu2,Uua)) ).

% ATP.lambda_76
tff(fact_8256_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ib(nat,fun(nat,A),Uu2),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(Uu2),Uua))) ) ).

% ATP.lambda_77
tff(fact_8257_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_gh(A,fun(nat,A),Uu2),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_78
tff(fact_8258_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_fa(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu2),Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uu2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_79
tff(fact_8259_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_gw(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu2),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_80
tff(fact_8260_ATP_Olambda__81,axiom,
    ! [A: $tType,Uu2: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_pt(set(set(A)),fun(set(set(A)),$o),Uu2),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu2)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_81
tff(fact_8261_ATP_Olambda__82,axiom,
    ! [A: $tType,Uu2: set(option(A)),Uua: option(A)] :
      ( aa(option(A),$o,aTP_Lamp_or(set(option(A)),fun(option(A),$o),Uu2),Uua)
    <=> ( member(option(A),Uua,Uu2)
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_82
tff(fact_8262_ATP_Olambda__83,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hy(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu2),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% ATP.lambda_83
tff(fact_8263_ATP_Olambda__84,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ey(real,fun(nat,real),Uu2),Uua) = divide_divide(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uua),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_84
tff(fact_8264_ATP_Olambda__85,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aij(nat,fun(nat,set(nat)),Uu2),Uua) = set_or3652927894154168847AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),Uua),Uu2) ).

% ATP.lambda_85
tff(fact_8265_ATP_Olambda__86,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_acd(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu2),Uua)),Uua) ).

% ATP.lambda_86
tff(fact_8266_ATP_Olambda__87,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hq(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),Uua)),Uu2) ).

% ATP.lambda_87
tff(fact_8267_ATP_Olambda__88,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hp(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),Uua)),Uua) ).

% ATP.lambda_88
tff(fact_8268_ATP_Olambda__89,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_adb(B,fun(A,set(product_prod(B,A))),Uu2),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),Uu2),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_89
tff(fact_8269_ATP_Olambda__90,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_aib(A,fun(B,set(product_prod(A,B))),Uu2),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),Uu2),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_90
tff(fact_8270_ATP_Olambda__91,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_bu(nat,fun(complex,$o),Uu2),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu2) = one_one(complex) ) ) ).

% ATP.lambda_91
tff(fact_8271_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_al(nat,fun(A,$o),Uu2),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu2) = one_one(A) ) ) ) ).

% ATP.lambda_92
tff(fact_8272_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_jl(A,fun(A,$o),Uu2),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu2) ) ) ) ).

% ATP.lambda_93
tff(fact_8273_ATP_Olambda__94,axiom,
    ! [A: $tType,Uu2: fun(set(A),$o),Uua: set(A)] :
      ( aa(set(A),$o,aa(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_acg(fun(set(A),$o),fun(set(A),$o)),Uu2),Uua)
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A10: set(A),A6: A] :
            ( ( Uua = aa(set(A),set(A),insert(A,A6),A10) )
            & aa(set(A),$o,Uu2,A10) ) ) ) ).

% ATP.lambda_94
tff(fact_8274_ATP_Olambda__95,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xl(real,fun(nat,real),Uu2),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)),divide_divide(real,Uu2,aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ).

% ATP.lambda_95
tff(fact_8275_ATP_Olambda__96,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_aad(real,fun(real,real),Uu2),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),Uu2),Uua)),divide_divide(real,one_one(real),Uua)) ).

% ATP.lambda_96
tff(fact_8276_ATP_Olambda__97,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_zj(real,fun(real,real),Uu2),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,Uu2,Uua)),Uua) ).

% ATP.lambda_97
tff(fact_8277_ATP_Olambda__98,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ck(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(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),Uu2),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_98
tff(fact_8278_ATP_Olambda__99,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_abr(set(product_prod(A,A)),fun(nat,$o),Uu2),Uua)
    <=> ( aa(nat,$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)),Uu2)) ) ) ).

% ATP.lambda_99
tff(fact_8279_ATP_Olambda__100,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_abw(nat,fun(nat,$o),Uu2),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,Uu2)) ) ) ).

% ATP.lambda_100
tff(fact_8280_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fv(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,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,Uu2,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_101
tff(fact_8281_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bz(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,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,Uu2,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_102
tff(fact_8282_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wv(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_103
tff(fact_8283_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bv(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_104
tff(fact_8284_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ke(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu2,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_105
tff(fact_8285_ATP_Olambda__106,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_kc(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu2,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_106
tff(fact_8286_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dr(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_107
tff(fact_8287_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cr(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_108
tff(fact_8288_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bk(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_109
tff(fact_8289_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dq(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_110
tff(fact_8290_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_by(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_111
tff(fact_8291_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bx(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_112
tff(fact_8292_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ww(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_113
tff(fact_8293_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cz(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_114
tff(fact_8294_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_my(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,$o,Uu2,Uua)
            & ! [Y5: A] :
                ( aa(A,$o,Uu2,Y5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),Uua) ) ) ) ) ).

% ATP.lambda_115
tff(fact_8295_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fm(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu2,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_116
tff(fact_8296_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gc(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu2,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_117
tff(fact_8297_ATP_Olambda__118,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_ue(fun(B,real),fun(B,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(B,real,Uu2,Uua)),zero_zero(real)) ) ).

% ATP.lambda_118
tff(fact_8298_ATP_Olambda__119,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_add(fun(B,A),fun(B,list(A)),Uu2),Uua) = aa(list(A),list(A),cons(A,aa(B,A,Uu2,Uua)),nil(A)) ).

% ATP.lambda_119
tff(fact_8299_ATP_Olambda__120,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_bg(fun(A,B),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,B,Uu2,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_120
tff(fact_8300_ATP_Olambda__121,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_fw(fun(A,B),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,B,Uu2,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_121
tff(fact_8301_ATP_Olambda__122,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_xv(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),Uu2)),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_122
tff(fact_8302_ATP_Olambda__123,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_xu(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_wk(fun(nat,real),fun(nat,real),Uu2)),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_123
tff(fact_8303_ATP_Olambda__124,axiom,
    ! [A: $tType,Uu2: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_ads(list(A),fun(list(A),list(list(A))),Uu2),Uua) = aa(list(A),list(list(A)),map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_acp(list(A),fun(A,list(A))),Uua)),Uu2) ).

% ATP.lambda_124
tff(fact_8304_ATP_Olambda__125,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_mp(nat,fun(nat,nat)),Uu2),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),Uu2),Uua))),Uu2) ).

% ATP.lambda_125
tff(fact_8305_ATP_Olambda__126,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_zo(fun(A,B),fun(A,real),Uu2),Uua) = divide_divide(real,real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua)),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_126
tff(fact_8306_ATP_Olambda__127,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_iq(A,fun(nat,A),Uu2),Uua) = 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),Uu2),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_127
tff(fact_8307_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ij(A,fun(nat,A),Uu2),Uua) = 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),Uu2),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_128
tff(fact_8308_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ig(A,fun(nat,A),Uu2),Uua) = 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),Uu2),Uua)) ) ).

% ATP.lambda_129
tff(fact_8309_ATP_Olambda__130,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_il(A,fun(nat,A),Uu2),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),Uu2),Uua)) ) ).

% ATP.lambda_130
tff(fact_8310_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ec(A,fun(nat,A),Uu2),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),Uu2),Uua)) ) ).

% ATP.lambda_131
tff(fact_8311_ATP_Olambda__132,axiom,
    ! [Uu2: num,Uua: num] : aa(num,int,aTP_Lamp_mx(num,fun(num,int),Uu2),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu2)),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),Uu2))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_132
tff(fact_8312_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_is(A,fun(nat,A),Uu2),Uua) = real_V8093663219630862766scaleR(A,cos_coeff(Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu2)),Uua)) ) ).

% ATP.lambda_133
tff(fact_8313_ATP_Olambda__134,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_qs(nat,fun(real,real),Uu2),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)),Uu2)) ).

% ATP.lambda_134
tff(fact_8314_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_im(A,fun(nat,A),Uu2),Uua) = real_V8093663219630862766scaleR(A,sin_coeff(Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua)) ) ).

% ATP.lambda_135
tff(fact_8315_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_in(A,fun(nat,A),Uu2),Uua) = real_V8093663219630862766scaleR(A,cos_coeff(Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua)) ) ).

% ATP.lambda_136
tff(fact_8316_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_xq(A,fun(nat,A),Uu2),Uua) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua)) ) ).

% ATP.lambda_137
tff(fact_8317_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_xp(A,fun(nat,A),Uu2),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),Uu2),Uua)) ) ).

% ATP.lambda_138
tff(fact_8318_ATP_Olambda__139,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ex(real,fun(nat,real),Uu2),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),Uu2),Uua)) ).

% ATP.lambda_139
tff(fact_8319_ATP_Olambda__140,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_es(real,fun(nat,real),Uu2),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),Uu2),Uua)) ).

% ATP.lambda_140
tff(fact_8320_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A,Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aap(A,fun(set(A),$o),Uu2),Uua)
        <=> ( topolo1002775350975398744n_open(A,Uua)
            & member(A,Uu2,Uua) ) ) ) ).

% ATP.lambda_141
tff(fact_8321_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mg(nat,fun(nat,A)),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Uu2)),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_142
tff(fact_8322_ATP_Olambda__143,axiom,
    ! [A: $tType,Uu2: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_adr(list(list(A)),fun(A,list(list(A))),Uu2),Uua) = aa(list(list(A)),list(list(A)),map(list(A),list(A),cons(A,Uua)),product_lists(A,Uu2)) ).

% ATP.lambda_143
tff(fact_8323_ATP_Olambda__144,axiom,
    ! [A: $tType,Uu2: list(A),Uua: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_qc(list(A),fun(list(A),$o)),Uu2),Uua)
    <=> ( aa(list(A),nat,size_size(list(A)),Uu2) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_144
tff(fact_8324_ATP_Olambda__145,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_kr(code_integer,fun(code_integer,num)),Uu2),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu2),
        $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_145
tff(fact_8325_ATP_Olambda__146,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_ks(code_integer,fun(code_integer,nat)),Uu2),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu2),
        $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_146
tff(fact_8326_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fb(nat,fun(nat,A),Uu2),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu2) ) ).

% ATP.lambda_147
tff(fact_8327_ATP_Olambda__148,axiom,
    ! [A: $tType,B: $tType,Uu2: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_adt(list(B),fun(A,list(product_prod(A,B))),Uu2),Uua) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua)),Uu2) ).

% ATP.lambda_148
tff(fact_8328_ATP_Olambda__149,axiom,
    ! [A: $tType,Uu2: set(nat),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_afs(set(nat),fun(product_prod(A,nat),$o),Uu2),Uua)
    <=> member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua),Uu2) ) ).

% ATP.lambda_149
tff(fact_8329_ATP_Olambda__150,axiom,
    ! [Uu2: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_qe(set(nat),fun(nat,$o),Uu2),Uua)
    <=> member(nat,aa(nat,nat,suc,Uua),Uu2) ) ).

% ATP.lambda_150
tff(fact_8330_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_cg(A,fun(A,product_prod(A,A))),Uu2),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),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_151
tff(fact_8331_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_ce(A,fun(A,product_prod(A,A))),Uu2),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),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_152
tff(fact_8332_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_fd(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),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_153
tff(fact_8333_ATP_Olambda__154,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ia(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu2),Uua)) ).

% ATP.lambda_154
tff(fact_8334_ATP_Olambda__155,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xn(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu2),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_155
tff(fact_8335_ATP_Olambda__156,axiom,
    ! [A: $tType,Uu2: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_aeo(fun(nat,A),fun(nat,product_prod(nat,A)),Uu2),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu2,Uua)) ).

% ATP.lambda_156
tff(fact_8336_ATP_Olambda__157,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_aco(fun(A,B),fun(A,product_prod(A,B)),Uu2),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu2,Uua)) ).

% ATP.lambda_157
tff(fact_8337_ATP_Olambda__158,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_jz(A,fun(nat,A),Uu2),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu2),Uua))) ) ).

% ATP.lambda_158
tff(fact_8338_ATP_Olambda__159,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_adz(fun(A,option(B)),fun(A,product_prod(A,B)),Uu2),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu2,Uua))) ).

% ATP.lambda_159
tff(fact_8339_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_xj(fun(nat,A),fun(nat,real),Uu2),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua))) ) ).

% ATP.lambda_160
tff(fact_8340_ATP_Olambda__161,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xk(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu2),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_161
tff(fact_8341_ATP_Olambda__162,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xc(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu2),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_162
tff(fact_8342_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ql(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_163
tff(fact_8343_ATP_Olambda__164,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_abv(set(product_prod(A,A)),fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu2)) ) ).

% ATP.lambda_164
tff(fact_8344_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_wt(A,fun(nat,A),Uu2),Uua) = divide_divide(A,Uu2,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_165
tff(fact_8345_ATP_Olambda__166,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_afl(nat,fun(list(A),$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu2),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_166
tff(fact_8346_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ha(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_167
tff(fact_8347_ATP_Olambda__168,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_kk(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_168
tff(fact_8348_ATP_Olambda__169,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_gm(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_169
tff(fact_8349_ATP_Olambda__170,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_afh(list(A),fun(A,$o),Uu2),Uua)
    <=> member(A,Uua,aa(list(A),set(A),set2(A),Uu2)) ) ).

% ATP.lambda_170
tff(fact_8350_ATP_Olambda__171,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ags(list(A),fun(A,$o),Uu2),Uua)
    <=> ( Uua = aa(list(A),A,hd(A),Uu2) ) ) ).

% ATP.lambda_171
tff(fact_8351_ATP_Olambda__172,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ahc(nat,fun(nat,$o)),Uu2),Uua)
    <=> ( Uua = aa(nat,nat,suc,Uu2) ) ) ).

% ATP.lambda_172
tff(fact_8352_ATP_Olambda__173,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ah(nat,fun(nat,$o)),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu2) ) ).

% ATP.lambda_173
tff(fact_8353_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_acu(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_174
tff(fact_8354_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ox(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_175
tff(fact_8355_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_fp(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_176
tff(fact_8356_ATP_Olambda__177,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_qq(nat,fun(nat,nat),Uu2),Uua) = modulo_modulo(nat,Uua,Uu2) ).

% ATP.lambda_177
tff(fact_8357_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_zd(A,fun(A,A),Uu2),Uua) = divide_divide(A,Uua,Uu2) ) ).

% ATP.lambda_178
tff(fact_8358_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_oc(A,fun(A,A),Uu2),Uua) = divide_divide(A,Uua,Uu2) ) ).

% ATP.lambda_179
tff(fact_8359_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_ack(A,fun(A,A),Uu2),Uua) = divide_divide(A,Uua,Uu2) ) ).

% ATP.lambda_180
tff(fact_8360_ATP_Olambda__181,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu2) ) ).

% ATP.lambda_181
tff(fact_8361_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_acv(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_182
tff(fact_8362_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_zz(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_183
tff(fact_8363_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_cs(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_184
tff(fact_8364_ATP_Olambda__185,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_wr(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu2) ).

% ATP.lambda_185
tff(fact_8365_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_zc(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_186
tff(fact_8366_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_aa(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_187
tff(fact_8367_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_og(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_188
tff(fact_8368_ATP_Olambda__189,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_nz(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu2) ).

% ATP.lambda_189
tff(fact_8369_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_ob(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu2) ) ).

% ATP.lambda_190
tff(fact_8370_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_of(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu2) ) ).

% ATP.lambda_191
tff(fact_8371_ATP_Olambda__192,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_rq(nat,fun(real,real),Uu2),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu2) ).

% ATP.lambda_192
tff(fact_8372_ATP_Olambda__193,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aen(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu2) ).

% ATP.lambda_193
tff(fact_8373_ATP_Olambda__194,axiom,
    ! [Uu2: int,Uua: int] : aa(int,int,aTP_Lamp_oo(int,fun(int,int),Uu2),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu2) ).

% ATP.lambda_194
tff(fact_8374_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_acl(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2) ) ).

% ATP.lambda_195
tff(fact_8375_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_oa(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2) ) ).

% ATP.lambda_196
tff(fact_8376_ATP_Olambda__197,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_rt(real,fun(real,real),Uu2),Uua) = powr(real,Uua,Uu2) ).

% ATP.lambda_197
tff(fact_8377_ATP_Olambda__198,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aw(nat,fun(nat,$o),Uu2),Uua)
    <=> dvd_dvd(nat,Uua,Uu2) ) ).

% ATP.lambda_198
tff(fact_8378_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_av(A,fun(A,$o),Uu2),Uua)
        <=> dvd_dvd(A,Uua,Uu2) ) ) ).

% ATP.lambda_199
tff(fact_8379_ATP_Olambda__200,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_ahz(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu2),Uua) = product_Sigma(A,B,Uua,Uu2) ).

% ATP.lambda_200
tff(fact_8380_ATP_Olambda__201,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_mf(nat,fun(nat,product_prod(nat,nat))),Uu2),Uua) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uu2) ).

% ATP.lambda_201
tff(fact_8381_ATP_Olambda__202,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_mc(B,fun(A,product_prod(A,B))),Uu2),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu2) ).

% ATP.lambda_202
tff(fact_8382_ATP_Olambda__203,axiom,
    ! [A: $tType,Uu2: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_aep(A,fun(nat,product_prod(nat,A)),Uu2),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),Uu2) ).

% ATP.lambda_203
tff(fact_8383_ATP_Olambda__204,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_mb(A,fun(B,product_prod(B,A))),Uu2),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu2) ).

% ATP.lambda_204
tff(fact_8384_ATP_Olambda__205,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ho(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,binomial(Uua),Uu2) ).

% ATP.lambda_205
tff(fact_8385_ATP_Olambda__206,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_acp(list(A),fun(A,list(A))),Uu2),Uua) = aa(list(A),list(A),cons(A,Uua),Uu2) ).

% ATP.lambda_206
tff(fact_8386_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: int,Uua: A] : aa(A,A,aTP_Lamp_agx(int,fun(A,A),Uu2),Uua) = power_int(A,Uua,Uu2) ) ).

% ATP.lambda_207
tff(fact_8387_ATP_Olambda__208,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xa(real,fun(nat,real),Uu2),Uua) = aa(real,real,root(Uua),Uu2) ).

% ATP.lambda_208
tff(fact_8388_ATP_Olambda__209,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_a(set(A),fun(A,$o),Uu2),Uua)
    <=> member(A,Uua,Uu2) ) ).

% ATP.lambda_209
tff(fact_8389_ATP_Olambda__210,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_abq(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu2),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),Uu2) ).

% ATP.lambda_210
tff(fact_8390_ATP_Olambda__211,axiom,
    ! [A: $tType,Uu2: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_qi(list(A),fun(nat,list(A)),Uu2),Uua) = drop(A,Uua,Uu2) ).

% ATP.lambda_211
tff(fact_8391_ATP_Olambda__212,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_afk(nat,fun(list(A),A),Uu2),Uua) = aa(nat,A,nth(A,Uua),Uu2) ).

% ATP.lambda_212
tff(fact_8392_ATP_Olambda__213,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_agq(A,fun(A,$o),Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_213
tff(fact_8393_ATP_Olambda__214,axiom,
    ! [A: $tType,Uu2: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_adv(A,fun(list(A),list(A))),Uu2),Uua) = aa(list(A),list(A),cons(A,Uu2),nil(A)) ).

% ATP.lambda_214
tff(fact_8394_ATP_Olambda__215,axiom,
    ! [A: $tType,Uu2: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_adw(A,fun(list(A),list(list(A)))),Uu2),Uua) = aa(list(list(A)),list(list(A)),cons(list(A),Uua),nil(list(A))) ).

% ATP.lambda_215
tff(fact_8395_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_yr(fun(A,real),fun(A,$o),Uu2),Uua)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu2,Uua)) ) ) ).

% ATP.lambda_216
tff(fact_8396_ATP_Olambda__217,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_yt(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_217
tff(fact_8397_ATP_Olambda__218,axiom,
    ! [B: $tType,A: $tType,Uu2: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_aia(set(product_prod(A,B)),fun(A,set(B)),Uu2),Uua) = image(product_prod(A,B),B,product_snd(A,B),Uu2) ).

% ATP.lambda_218
tff(fact_8398_ATP_Olambda__219,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_dh(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(nat,real,Uu2,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_219
tff(fact_8399_ATP_Olambda__220,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_dg(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(nat,real,Uu2,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_220
tff(fact_8400_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_zn(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,Uu2,divide_divide(A,one_one(A),Uua)) ) ).

% ATP.lambda_221
tff(fact_8401_ATP_Olambda__222,axiom,
    ! [A: $tType,Uu2: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_afq(fun(nat,$o),fun(product_prod(A,nat),$o),Uu2),Uua)
    <=> aa(nat,$o,Uu2,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_222
tff(fact_8402_ATP_Olambda__223,axiom,
    ! [A: $tType,Uu2: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_afr(fun(nat,$o),fun(product_prod(A,nat),$o),Uu2),Uua)
    <=> aa(nat,$o,Uu2,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)) ) ).

% ATP.lambda_223
tff(fact_8403_ATP_Olambda__224,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_xy(fun(nat,$o),fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_224
tff(fact_8404_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dp(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_225
tff(fact_8405_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wo(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_226
tff(fact_8406_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ef(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_227
tff(fact_8407_ATP_Olambda__228,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fl(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_228
tff(fact_8408_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bp(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_229
tff(fact_8409_ATP_Olambda__230,axiom,
    ! [A: $tType,Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_xw(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_230
tff(fact_8410_ATP_Olambda__231,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_pc(A,fun(option(A),option(A))),Uu2),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu2,aa(A,fun(A,A),ord_min(A),Uu2),Uua)) ) ).

% ATP.lambda_231
tff(fact_8411_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_pe(A,fun(option(A),option(A))),Uu2),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu2,aa(A,fun(A,A),ord_max(A),Uu2),Uua)) ) ).

% ATP.lambda_232
tff(fact_8412_ATP_Olambda__233,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_acw(A,fun(option(A),option(A))),Uu2),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu2,aa(A,fun(A,A),sup_sup(A),Uu2),Uua)) ) ).

% ATP.lambda_233
tff(fact_8413_ATP_Olambda__234,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_pd(A,fun(option(A),option(A))),Uu2),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu2,aa(A,fun(A,A),inf_inf(A),Uu2),Uua)) ) ).

% ATP.lambda_234
tff(fact_8414_ATP_Olambda__235,axiom,
    ! [Uu2: nat,Uua: num] : aa(num,option(num),aTP_Lamp_nf(nat,fun(num,option(num)),Uu2),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu2,Uua))) ).

% ATP.lambda_235
tff(fact_8415_ATP_Olambda__236,axiom,
    ! [Uu2: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_nc(num,fun(nat,option(num)),Uu2),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uua,Uu2))) ).

% ATP.lambda_236
tff(fact_8416_ATP_Olambda__237,axiom,
    ! [A: $tType,Uu2: 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_qu(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu2),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_qt(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu2),Uua)) ).

% ATP.lambda_237
tff(fact_8417_ATP_Olambda__238,axiom,
    ! [Uu2: 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_mo(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu2),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_mn(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu2),Uua)) ).

% ATP.lambda_238
tff(fact_8418_ATP_Olambda__239,axiom,
    ! [Uu2: 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_mm(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu2),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_ml(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu2),Uua)) ).

% ATP.lambda_239
tff(fact_8419_ATP_Olambda__240,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_mk(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu2),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_mj(nat,fun(nat,fun(nat,fun(nat,$o))),Uu2),Uua)) ).

% ATP.lambda_240
tff(fact_8420_ATP_Olambda__241,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_mi(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu2),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_mh(nat,fun(nat,fun(nat,fun(nat,$o))),Uu2),Uua)) ).

% ATP.lambda_241
tff(fact_8421_ATP_Olambda__242,axiom,
    ! [Uu2: 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_me(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu2),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_md(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu2),Uua)) ).

% ATP.lambda_242
tff(fact_8422_ATP_Olambda__243,axiom,
    ! [Uu2: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_sc(fun(nat,real),fun(real,real),Uu2),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_sb(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua)) ).

% ATP.lambda_243
tff(fact_8423_ATP_Olambda__244,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_rr(fun(nat,A),fun(A,A),Uu2),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_ep(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua)) ) ).

% ATP.lambda_244
tff(fact_8424_ATP_Olambda__245,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,$o)),Uua: B] : aa(B,set(A),aTP_Lamp_qp(fun(A,fun(B,$o)),fun(B,set(A)),Uu2),Uua) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_qo(fun(A,fun(B,$o)),fun(B,fun(A,$o)),Uu2),Uua)) ).

% ATP.lambda_245
tff(fact_8425_ATP_Olambda__246,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_ja(nat,fun(nat,complex),Uu2),Uua) = cis(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),Uu2))) ).

% ATP.lambda_246
tff(fact_8426_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_te(A,fun(A,A),Uu2),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),Uu2)),Uua)),aa(A,A,inverse_inverse(A),Uu2))) ) ).

% ATP.lambda_247
tff(fact_8427_ATP_Olambda__248,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_afm(fun(B,option(A)),fun(B,$o),Uu2),Uua)
    <=> ( aa(B,option(A),Uu2,Uua) != none(A) ) ) ).

% ATP.lambda_248
tff(fact_8428_ATP_Olambda__249,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,option(B)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ady(fun(A,option(B)),fun(A,$o),Uu2),Uua)
    <=> ( aa(A,option(B),Uu2,Uua) != none(B) ) ) ).

% ATP.lambda_249
tff(fact_8429_ATP_Olambda__250,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_aic(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu2),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),image(B,set(product_prod(A,B)),aTP_Lamp_aib(A,fun(B,set(product_prod(A,B))),Uua),aa(A,set(B),Uu2,Uua))) ).

% ATP.lambda_250
tff(fact_8430_ATP_Olambda__251,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_aed(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu2),Uua) = concat(product_prod(A,B),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_aec(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu2)) ).

% ATP.lambda_251
tff(fact_8431_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,real,aTP_Lamp_ih(A,fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(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),Uu2),Uua))) ) ).

% ATP.lambda_252
tff(fact_8432_ATP_Olambda__253,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_bh(nat,fun(nat,$o),Uu2),Uua)
    <=> ~ dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),divide_divide(nat,Uu2,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_253
tff(fact_8433_ATP_Olambda__254,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ir(A,fun(nat,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),real_V8093663219630862766scaleR(A,sin_coeff(Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu2)),Uua))) ) ).

% ATP.lambda_254
tff(fact_8434_ATP_Olambda__255,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,real,aTP_Lamp_io(A,fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,real_V8093663219630862766scaleR(A,sin_coeff(Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua))) ) ).

% ATP.lambda_255
tff(fact_8435_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,real,aTP_Lamp_ip(A,fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,real_V8093663219630862766scaleR(A,cos_coeff(Uua),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua))) ) ).

% ATP.lambda_256
tff(fact_8436_ATP_Olambda__257,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fc(A,fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_257
tff(fact_8437_ATP_Olambda__258,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fe(A,fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_258
tff(fact_8438_ATP_Olambda__259,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_afg(list(A),fun(A,$o),Uu2),Uua)
    <=> ~ member(A,Uua,aa(list(A),set(A),set2(A),Uu2)) ) ).

% ATP.lambda_259
tff(fact_8439_ATP_Olambda__260,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aii(nat,fun(nat,set(nat)),Uu2),Uua) = set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),Uua)) ).

% ATP.lambda_260
tff(fact_8440_ATP_Olambda__261,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xi(real,fun(nat,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uua)) ).

% ATP.lambda_261
tff(fact_8441_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_rn(A,fun(A,A),Uu2),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2)) ) ).

% ATP.lambda_262
tff(fact_8442_ATP_Olambda__263,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: A,Uua: A] : aa(A,filter(A),aTP_Lamp_aan(A,fun(A,filter(A)),Uu2),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uu2,Uua)) ) ).

% ATP.lambda_263
tff(fact_8443_ATP_Olambda__264,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: A,Uua: A] : aa(A,filter(A),aTP_Lamp_aam(A,fun(A,filter(A)),Uu2),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uua,Uu2)) ) ).

% ATP.lambda_264
tff(fact_8444_ATP_Olambda__265,axiom,
    ! [B: $tType,A: $tType,Uu2: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_adc(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu2),Uua) = insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu2),Uua)) ).

% ATP.lambda_265
tff(fact_8445_ATP_Olambda__266,axiom,
    ! [A: $tType,B: $tType,Uu2: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_aid(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu2),Uua) = insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu2),Uua)) ).

% ATP.lambda_266
tff(fact_8446_ATP_Olambda__267,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_oy(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uu2),Uua)) ).

% ATP.lambda_267
tff(fact_8447_ATP_Olambda__268,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_oz(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uua),Uu2)) ).

% ATP.lambda_268
tff(fact_8448_ATP_Olambda__269,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ll(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu2),Uua)) ).

% ATP.lambda_269
tff(fact_8449_ATP_Olambda__270,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_lk(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu2)) ).

% ATP.lambda_270
tff(fact_8450_ATP_Olambda__271,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_aey(A,fun(A,$o),Uu2),Uua)
    <=> ( Uu2 != Uua ) ) ).

% ATP.lambda_271
tff(fact_8451_ATP_Olambda__272,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_aff(A,fun(A,$o),Uu2),Uua)
    <=> ( Uua != Uu2 ) ) ).

% ATP.lambda_272
tff(fact_8452_ATP_Olambda__273,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_dv(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_273
tff(fact_8453_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_gq(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_274
tff(fact_8454_ATP_Olambda__275,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_wy(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_275
tff(fact_8455_ATP_Olambda__276,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_tf(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_276
tff(fact_8456_ATP_Olambda__277,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_qw(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_277
tff(fact_8457_ATP_Olambda__278,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tn(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_278
tff(fact_8458_ATP_Olambda__279,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fq(fun(nat,A),fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_279
tff(fact_8459_ATP_Olambda__280,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gt(fun(nat,A),fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_280
tff(fact_8460_ATP_Olambda__281,axiom,
    ! [Uu2: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_abf(fun(real,real),fun(real,real),Uu2),Uua) = aa(real,real,artanh(real),aa(real,real,Uu2,Uua)) ).

% ATP.lambda_281
tff(fact_8461_ATP_Olambda__282,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zx(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,artanh(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_282
tff(fact_8462_ATP_Olambda__283,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uu(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,artanh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_283
tff(fact_8463_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tz(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arctan,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_284
tff(fact_8464_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sm(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcsin,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_285
tff(fact_8465_ATP_Olambda__286,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abe(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcsin,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_286
tff(fact_8466_ATP_Olambda__287,axiom,
    ! [Uu2: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_abb(fun(real,real),fun(real,real),Uu2),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu2,Uua)) ).

% ATP.lambda_287
tff(fact_8467_ATP_Olambda__288,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ys(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_288
tff(fact_8468_ATP_Olambda__289,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vv(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_289
tff(fact_8469_ATP_Olambda__290,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_so(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arccos,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_290
tff(fact_8470_ATP_Olambda__291,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_abd(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arccos,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_291
tff(fact_8471_ATP_Olambda__292,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_dw(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_292
tff(fact_8472_ATP_Olambda__293,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_bj(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_293
tff(fact_8473_ATP_Olambda__294,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rz(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,tanh(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_294
tff(fact_8474_ATP_Olambda__295,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rf(fun(A,A),fun(A,A),Uu2),Uua) = sinh(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_295
tff(fact_8475_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rg(fun(A,A),fun(A,A),Uu2),Uua) = cosh(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_296
tff(fact_8476_ATP_Olambda__297,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ub(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,tan(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_297
tff(fact_8477_ATP_Olambda__298,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ta(fun(A,real),fun(A,real),Uu2),Uua) = sin(real,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_298
tff(fact_8478_ATP_Olambda__299,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rj(fun(A,A),fun(A,A),Uu2),Uua) = sin(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_299
tff(fact_8479_ATP_Olambda__300,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sy(fun(A,real),fun(A,real),Uu2),Uua) = exp(real,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_300
tff(fact_8480_ATP_Olambda__301,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ri(fun(A,A),fun(A,A),Uu2),Uua) = exp(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_301
tff(fact_8481_ATP_Olambda__302,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_xt(fun(nat,real),fun(nat,real),Uu2),Uua) = cos(real,aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_302
tff(fact_8482_ATP_Olambda__303,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tj(fun(A,real),fun(A,real),Uu2),Uua) = cos(real,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_303
tff(fact_8483_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_rm(fun(A,A),fun(A,A),Uu2),Uua) = cos(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_304
tff(fact_8484_ATP_Olambda__305,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_nu(fun(B,A),fun(B,option(A)),Uu2),Uua) = aa(A,option(A),some(A),aa(B,A,Uu2,Uua)) ).

% ATP.lambda_305
tff(fact_8485_ATP_Olambda__306,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_adx(fun(A,B),fun(A,option(B)),Uu2),Uua) = aa(B,option(B),some(B),aa(A,B,Uu2,Uua)) ).

% ATP.lambda_306
tff(fact_8486_ATP_Olambda__307,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_ade(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu2),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu2,Uua)) ).

% ATP.lambda_307
tff(fact_8487_ATP_Olambda__308,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_ado(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu2),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uua)) ).

% ATP.lambda_308
tff(fact_8488_ATP_Olambda__309,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_pl(fun(A,set(B)),fun(A,nat),Uu2),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu2,Uua)) ).

% ATP.lambda_309
tff(fact_8489_ATP_Olambda__310,axiom,
    ! [Uu2: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_ry(fun(real,fun(nat,real)),fun(real,real),Uu2),Uua) = suminf(real,aa(real,fun(nat,real),Uu2,Uua)) ).

% ATP.lambda_310
tff(fact_8490_ATP_Olambda__311,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu2: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_we(fun(A,fun(nat,B)),fun(A,B),Uu2),Uua) = suminf(B,aa(A,fun(nat,B),Uu2,Uua)) ) ).

% ATP.lambda_311
tff(fact_8491_ATP_Olambda__312,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_agt(fun(B,list(A)),fun(B,set(A)),Uu2),Uua) = aa(list(A),set(A),set2(A),aa(B,list(A),Uu2,Uua)) ).

% ATP.lambda_312
tff(fact_8492_ATP_Olambda__313,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_tx(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,sqrt,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_313
tff(fact_8493_ATP_Olambda__314,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,fun(B,$o)),Uua: A] : aa(A,set(B),aTP_Lamp_aht(fun(A,fun(B,$o)),fun(A,set(B)),Uu2),Uua) = aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),Uu2,Uua)) ).

% ATP.lambda_314
tff(fact_8494_ATP_Olambda__315,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_kz(fun(A,nat),fun(A,nat),Uu2),Uua) = aa(nat,nat,suc,aa(A,nat,Uu2,Uua)) ).

% ATP.lambda_315
tff(fact_8495_ATP_Olambda__316,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aex(fun(A,$o),fun(A,$o),Uu2),Uua)
    <=> ~ aa(A,$o,Uu2,Uua) ) ).

% ATP.lambda_316
tff(fact_8496_ATP_Olambda__317,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_qn(fun(A,fun(B,$o)),fun(A,$o),Uu2),Uua)
    <=> ? [X_12: B] : aa(B,$o,aa(A,fun(B,$o),Uu2,Uua),X_12) ) ).

% ATP.lambda_317
tff(fact_8497_ATP_Olambda__318,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,fun(B,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_zt(fun(A,fun(B,$o)),fun(B,$o),Uu2),Uua)
        <=> ? [I3: A] : aa(B,$o,aa(A,fun(B,$o),Uu2,I3),Uua) ) ) ).

% ATP.lambda_318
tff(fact_8498_ATP_Olambda__319,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ac(list(A),fun(A,$o),Uu2),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu2),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu2)) ) ) ).

% ATP.lambda_319
tff(fact_8499_ATP_Olambda__320,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_pz(set(set(A)),fun(set(A),$o),Uu2),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image(set(A),A,F5,Uu2) )
              & ! [X3: set(A)] :
                  ( member(set(A),X3,Uu2)
                 => member(A,aa(set(A),A,F5,X3),X3) ) ) ) ) ).

% ATP.lambda_320
tff(fact_8500_ATP_Olambda__321,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_qa(set(set(A)),fun(set(A),$o),Uu2),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image(set(A),A,F5,Uu2) )
              & ! [X3: set(A)] :
                  ( member(set(A),X3,Uu2)
                 => member(A,aa(set(A),A,F5,X3),X3) ) ) ) ) ).

% ATP.lambda_321
tff(fact_8501_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu2: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_qb(set(set(A)),fun(set(A),$o),Uu2),Uua)
        <=> ? [F5: fun(set(A),A)] :
              ( ( Uua = image(set(A),A,F5,Uu2) )
              & ! [X3: set(A)] :
                  ( member(set(A),X3,Uu2)
                 => member(A,aa(set(A),A,F5,X3),X3) ) ) ) ) ).

% ATP.lambda_322
tff(fact_8502_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_pv(set(A),fun(A,$o),Uu2),Uua)
        <=> ! [X3: A] :
              ( member(A,X3,Uu2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X3) ) ) ) ).

% ATP.lambda_323
tff(fact_8503_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_pw(set(A),fun(A,$o),Uu2),Uua)
        <=> ! [X3: A] :
              ( member(A,X3,Uu2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Uua) ) ) ) ).

% ATP.lambda_324
tff(fact_8504_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_yd(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y5)
             => aa(A,$o,Uu2,Y5) ) ) ) ).

% ATP.lambda_325
tff(fact_8505_ATP_Olambda__326,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,option(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_acq(fun(B,option(A)),fun(A,$o),Uu2),Uua)
    <=> ? [A6: B] : aa(B,option(A),Uu2,A6) = aa(A,option(A),some(A),Uua) ) ).

% ATP.lambda_326
tff(fact_8506_ATP_Olambda__327,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_qm(fun(B,A),fun(A,$o),Uu2),Uua)
    <=> ? [X3: B] : Uua = aa(B,A,Uu2,X3) ) ).

% ATP.lambda_327
tff(fact_8507_ATP_Olambda__328,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_aei(fun(A,option(B)),fun(product_prod(A,B),$o),Uu2),Uua)
    <=> ? [A6: A,B6: B] :
          ( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6) )
          & ( aa(A,option(B),Uu2,A6) = aa(B,option(B),some(B),B6) ) ) ) ).

% ATP.lambda_328
tff(fact_8508_ATP_Olambda__329,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aih(nat,fun(nat,set(nat)),Uu2),Uua) = set_ord_atMost(nat,Uu2) ).

% ATP.lambda_329
tff(fact_8509_ATP_Olambda__330,axiom,
    ! [A: $tType,B: $tType,Uu2: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_ahg(set(B),fun(A,set(B)),Uu2),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu2) ).

% ATP.lambda_330
tff(fact_8510_ATP_Olambda__331,axiom,
    ! [A: $tType,Uu2: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_agr(A,fun(list(A),option(A))),Uu2),Uua) = aa(A,option(A),some(A),Uu2) ).

% ATP.lambda_331
tff(fact_8511_ATP_Olambda__332,axiom,
    ! [A: $tType,B: $tType,Uu2: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_ahk(list(B),fun(A,set(B)),Uu2),Uua) = aa(list(B),set(B),set2(B),Uu2) ).

% ATP.lambda_332
tff(fact_8512_ATP_Olambda__333,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,$o),Uua: A] : aa(A,set(B),aTP_Lamp_ahe(fun(B,$o),fun(A,set(B)),Uu2),Uua) = aa(fun(B,$o),set(B),collect(B),Uu2) ).

% ATP.lambda_333
tff(fact_8513_ATP_Olambda__334,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_em(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub),aa(nat,real,Uua,divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,Uu2,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_334
tff(fact_8514_ATP_Olambda__335,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_df(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = $ite(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub),aa(nat,real,Uu2,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_335
tff(fact_8515_ATP_Olambda__336,axiom,
    ! [Uu2: 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_ko(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),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),Uu2)),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),Uu2))),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_336
tff(fact_8516_ATP_Olambda__337,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_cb(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu2),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu2)),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),Uu2))),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_337
tff(fact_8517_ATP_Olambda__338,axiom,
    ! [Uu2: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_cc(num,fun(int,fun(int,product_prod(int,int))),Uu2),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu2)),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),Uu2))),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_338
tff(fact_8518_ATP_Olambda__339,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: num,Uua: A,Uub: A] :
          aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_bi(num,fun(A,fun(A,product_prod(A,A))),Uu2),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu2)),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),Uu2))),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_339
tff(fact_8519_ATP_Olambda__340,axiom,
    ! [A: $tType,Uu2: A,Uua: A,Uub: list(A)] :
      aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ach(A,fun(A,fun(list(A),list(A))),Uu2),Uua),Uub) = $ite(Uu2 = Uua,aa(list(A),list(A),cons(A,Uua),Uub),aa(list(A),list(A),cons(A,Uu2),aa(list(A),list(A),cons(A,Uua),Uub))) ).

% ATP.lambda_340
tff(fact_8520_ATP_Olambda__341,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_nm(fun(A,B),fun(set(A),fun(A,B)),Uu2),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu2,Uub),one_one(B)) ) ).

% ATP.lambda_341
tff(fact_8521_ATP_Olambda__342,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_fh(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_342
tff(fact_8522_ATP_Olambda__343,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_fg(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_343
tff(fact_8523_ATP_Olambda__344,axiom,
    ! [Uu2: 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_lt(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),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),Uu2)),Uub))) ).

% ATP.lambda_344
tff(fact_8524_ATP_Olambda__345,axiom,
    ! [Uu2: 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_lw(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),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),Uu2)),Uub))) ).

% ATP.lambda_345
tff(fact_8525_ATP_Olambda__346,axiom,
    ! [Uu2: 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_ls(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),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),Uu2),Uub))) ).

% ATP.lambda_346
tff(fact_8526_ATP_Olambda__347,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_fs(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu2,Uub),one_one(B)) ) ).

% ATP.lambda_347
tff(fact_8527_ATP_Olambda__348,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,fun(A,A)),Uua: fun(B,$o),Uub: B] :
      aa(B,fun(A,A),aa(fun(B,$o),fun(B,fun(A,A)),aTP_Lamp_afa(fun(B,fun(A,A)),fun(fun(B,$o),fun(B,fun(A,A))),Uu2),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,fun(A,A),Uu2,Uub),id(A)) ).

% ATP.lambda_348
tff(fact_8528_ATP_Olambda__349,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_afu(fun(B,A),fun(fun(B,$o),fun(B,option(A))),Uu2),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu2,Uub)),none(A)) ).

% ATP.lambda_349
tff(fact_8529_ATP_Olambda__350,axiom,
    ! [A: $tType,B: $tType,Uu2: 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_aie(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu2),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_aid(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu2) ).

% ATP.lambda_350
tff(fact_8530_ATP_Olambda__351,axiom,
    ! [B: $tType,A: $tType,Uu2: 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_aeq(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu2),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_adc(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu2) ).

% ATP.lambda_351
tff(fact_8531_ATP_Olambda__352,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,option(B)),Uua: A,Uub: B] : aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_afw(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu2),Uua),Uub) = fun_upd(A,option(B),Uu2,Uua,aa(B,option(B),some(B),Uub)) ).

% ATP.lambda_352
tff(fact_8532_ATP_Olambda__353,axiom,
    ! [A: $tType,Uu2: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_acr(fun(A,A),fun(A,fun(nat,A)),Uu2),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),Uu2),Uua) ).

% ATP.lambda_353
tff(fact_8533_ATP_Olambda__354,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Uu2: 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_lq(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),Uu2),Uua),Uub) = aa(B,C,aa(D,fun(B,C),Uu2,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_354
tff(fact_8534_ATP_Olambda__355,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ht(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_355
tff(fact_8535_ATP_Olambda__356,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_356
tff(fact_8536_ATP_Olambda__357,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,fun(B,$o)),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ye(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_357
tff(fact_8537_ATP_Olambda__358,axiom,
    ! [Uu2: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_rx(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu2,Uub),Uua) ).

% ATP.lambda_358
tff(fact_8538_ATP_Olambda__359,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_359
tff(fact_8539_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gd(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_360
tff(fact_8540_ATP_Olambda__361,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,fun(A,$o)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ak(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> aa(A,$o,aa(B,fun(A,$o),Uu2,Uub),Uua) ) ).

% ATP.lambda_361
tff(fact_8541_ATP_Olambda__362,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_us(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_362
tff(fact_8542_ATP_Olambda__363,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,fun(A,A)),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_agk(fun(B,fun(A,A)),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(B,fun(A,A),Uu2,Uub),Uua) ).

% ATP.lambda_363
tff(fact_8543_ATP_Olambda__364,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_tr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_364
tff(fact_8544_ATP_Olambda__365,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,$o)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_qo(fun(A,fun(B,$o)),fun(B,fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu2,Uub),Uua) ) ).

% ATP.lambda_365
tff(fact_8545_ATP_Olambda__366,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,A)),Uua: B,Uub: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_agl(fun(A,fun(B,A)),fun(B,fun(A,A)),Uu2),Uua),Uub) = aa(B,A,aa(A,fun(B,A),Uu2,Uub),Uua) ).

% ATP.lambda_366
tff(fact_8546_ATP_Olambda__367,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agm(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu2,Uub),Uua) ) ).

% ATP.lambda_367
tff(fact_8547_ATP_Olambda__368,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_iu(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_it(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_368
tff(fact_8548_ATP_Olambda__369,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_if(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ie(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_369
tff(fact_8549_ATP_Olambda__370,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_id(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ic(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_370
tff(fact_8550_ATP_Olambda__371,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gs(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),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_gr(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_371
tff(fact_8551_ATP_Olambda__372,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat,Uua: A,Uub: nat] :
          aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eh(nat,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu2,one_one(A),zero_zero(A))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_372
tff(fact_8552_ATP_Olambda__373,axiom,
    ! [Uu2: 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_lm(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu2),Uua),Uub) = aa($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)),Uu2),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_373
tff(fact_8553_ATP_Olambda__374,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fo(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),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_fn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_374
tff(fact_8554_ATP_Olambda__375,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ge(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gd(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_375
tff(fact_8555_ATP_Olambda__376,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_wg(fun(nat,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_ct(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_376
tff(fact_8556_ATP_Olambda__377,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jx(rat,fun(int,fun(int,product_prod(int,int))),Uu2),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_jw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_377
tff(fact_8557_ATP_Olambda__378,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jv(rat,fun(int,fun(int,product_prod(int,int))),Uu2),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_ju(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_378
tff(fact_8558_ATP_Olambda__379,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jt(rat,fun(int,fun(int,product_prod(int,int))),Uu2),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_js(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_379
tff(fact_8559_ATP_Olambda__380,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jr(rat,fun(int,fun(int,product_prod(int,int))),Uu2),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_jq(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu2)) ).

% ATP.lambda_380
tff(fact_8560_ATP_Olambda__381,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_jp(rat,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> aa(product_prod(int,int),$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_jo(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_381
tff(fact_8561_ATP_Olambda__382,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_jn(rat,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> aa(product_prod(int,int),$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_jm(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_382
tff(fact_8562_ATP_Olambda__383,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ts(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),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_tr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_383
tff(fact_8563_ATP_Olambda__384,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sd(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu2,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_384
tff(fact_8564_ATP_Olambda__385,axiom,
    ! [Uu2: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_se(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(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),Uu2),Uub)) ).

% ATP.lambda_385
tff(fact_8565_ATP_Olambda__386,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu2: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_eu(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(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),Uu2),Uub)) ) ).

% ATP.lambda_386
tff(fact_8566_ATP_Olambda__387,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_gl(fun(nat,A),fun(nat,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_387
tff(fact_8567_ATP_Olambda__388,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_wb(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu2,Uua)),Uub) ) ).

% ATP.lambda_388
tff(fact_8568_ATP_Olambda__389,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_vx(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu2,Uua)),Uub) ) ).

% ATP.lambda_389
tff(fact_8569_ATP_Olambda__390,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bl(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_390
tff(fact_8570_ATP_Olambda__391,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ul(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu2,Uub)),aa(A,A,Uu2,Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_391
tff(fact_8571_ATP_Olambda__392,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_vy(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu2,Uub)),aa(A,A,Uu2,Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_392
tff(fact_8572_ATP_Olambda__393,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sa(fun(nat,real),fun(real,fun(nat,real)),Uu2),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,Uu2,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_393
tff(fact_8573_ATP_Olambda__394,axiom,
    ! [Uu2: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ev(real,fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(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),Uu2),Uub)) ).

% ATP.lambda_394
tff(fact_8574_ATP_Olambda__395,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_agb(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),one_one(nat)) )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_395
tff(fact_8575_ATP_Olambda__396,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eo(fun(nat,A),fun(A,fun(nat,A)),Uu2),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,Uu2,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_396
tff(fact_8576_ATP_Olambda__397,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_ot(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu2),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) )
          & 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,Uu2)) ) ) ) ).

% ATP.lambda_397
tff(fact_8577_ATP_Olambda__398,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_qh(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X3: A,Y5: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),cons(A,X3),Xs5)) )
            & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),cons(A,Y5),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y5),Uu2) ) ) ) ).

% ATP.lambda_398
tff(fact_8578_ATP_Olambda__399,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_agc(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu2 )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_399
tff(fact_8579_ATP_Olambda__400,axiom,
    ! [A: $tType,Uu2: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_lg(nat,fun(set(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu2 )
        & 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_400
tff(fact_8580_ATP_Olambda__401,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_lf(set(A),fun(nat,fun(list(A),$o)),Uu2),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)),Uu2) ) ) ).

% ATP.lambda_401
tff(fact_8581_ATP_Olambda__402,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_aha(set(A),fun(nat,fun(list(A),$o)),Uu2),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)),Uu2)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_402
tff(fact_8582_ATP_Olambda__403,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_kp(nat,fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu2 )
        & 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_403
tff(fact_8583_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ag(set(A),fun(nat,fun(list(A),$o)),Uu2),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)),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_404
tff(fact_8584_ATP_Olambda__405,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_af(set(A),fun(nat,fun(list(A),$o)),Uu2),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)),Uu2)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_405
tff(fact_8585_ATP_Olambda__406,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_aga(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu2 )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_406
tff(fact_8586_ATP_Olambda__407,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ru(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu2)),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_407
tff(fact_8587_ATP_Olambda__408,axiom,
    ! [Uu2: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_kx(set(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_408
tff(fact_8588_ATP_Olambda__409,axiom,
    ! [A: $tType,Uu2: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_aft(set(nat),fun(nat,fun(product_prod(A,nat),$o)),Uu2),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua),Uu2) ) ).

% ATP.lambda_409
tff(fact_8589_ATP_Olambda__410,axiom,
    ! [A: $tType,Uu2: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_no(list(A),fun(set(nat),fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu2))
        & member(nat,Uub,Uua) ) ) ).

% ATP.lambda_410
tff(fact_8590_ATP_Olambda__411,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_afi(fun(A,$o),fun(list(A),fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua))
        & aa(A,$o,Uu2,aa(nat,A,nth(A,Uua),Uub)) ) ) ).

% ATP.lambda_411
tff(fact_8591_ATP_Olambda__412,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_agp(list(A),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uu2))
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_412
tff(fact_8592_ATP_Olambda__413,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_aew(fun(A,$o),fun(list(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uua))
        & aa(A,$o,Uu2,Uub) ) ) ).

% ATP.lambda_413
tff(fact_8593_ATP_Olambda__414,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(A,fun(product_prod(A,B),$o),aTP_Lamp_aej(fun(A,option(B)),fun(A,fun(product_prod(A,B),$o)),Uu2),Uua),Uub)
    <=> ( member(product_prod(A,B),Uub,graph(A,B,Uu2))
        & ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).

% ATP.lambda_414
tff(fact_8594_ATP_Olambda__415,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kj(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu2),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_415
tff(fact_8595_ATP_Olambda__416,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_air(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu2))
        & ! [X3: A] :
            ( member(A,X3,Uua)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X3),Uu2) ) ) ) ).

% ATP.lambda_416
tff(fact_8596_ATP_Olambda__417,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ais(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu2))
        & ! [X3: A] :
            ( member(A,X3,Uua)
           => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Uub),Uu2) ) ) ) ).

% ATP.lambda_417
tff(fact_8597_ATP_Olambda__418,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aiq(set(product_prod(A,A)),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,field2(A,Uu2))
        & ! [X3: A] :
            ( member(A,X3,Uua)
           => ( ( Uub != X3 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X3),Uu2) ) ) ) ) ).

% ATP.lambda_418
tff(fact_8598_ATP_Olambda__419,axiom,
    ! [A: $tType,Uu2: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_qd(list(A),fun(set(nat),fun(nat,$o)),Uu2),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu2)),Uua) ) ).

% ATP.lambda_419
tff(fact_8599_ATP_Olambda__420,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ait(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uub) ) ) ).

% ATP.lambda_420
tff(fact_8600_ATP_Olambda__421,axiom,
    ! [Uu2: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ky(set(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,Uub,Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_421
tff(fact_8601_ATP_Olambda__422,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_en(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu2),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_422
tff(fact_8602_ATP_Olambda__423,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_eq(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu2),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_423
tff(fact_8603_ATP_Olambda__424,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_er(A,fun(fun(nat,A),fun(nat,A)),Uu2),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),Uu2),Uub)) ) ).

% ATP.lambda_424
tff(fact_8604_ATP_Olambda__425,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hr(nat,fun(nat,fun(nat,nat)),Uu2),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),Uu2),Uub)) ).

% ATP.lambda_425
tff(fact_8605_ATP_Olambda__426,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_pg(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( dvd_dvd(nat,Uub,Uua)
        & dvd_dvd(nat,Uub,Uu2) ) ) ).

% ATP.lambda_426
tff(fact_8606_ATP_Olambda__427,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aip(set(A),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uua,Uu2)
        & member(A,Uub,Uu2) ) ) ).

% ATP.lambda_427
tff(fact_8607_ATP_Olambda__428,axiom,
    ! [A: $tType,Uu2: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_afv(list(A),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu2),Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_428
tff(fact_8608_ATP_Olambda__429,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_agd(fun(A,nat),fun(list(A),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uua),Uub)),aa(A,nat,Uu2,Uub)) ).

% ATP.lambda_429
tff(fact_8609_ATP_Olambda__430,axiom,
    ! [A: $tType,Uu2: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_fr(set(A),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_430
tff(fact_8610_ATP_Olambda__431,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(A,set(B)),Uub: A] :
      ( aa(A,$o,aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_ahq(set(A),fun(fun(A,set(B)),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_431
tff(fact_8611_ATP_Olambda__432,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_at(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_432
tff(fact_8612_ATP_Olambda__433,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_jy(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_433
tff(fact_8613_ATP_Olambda__434,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ar(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_434
tff(fact_8614_ATP_Olambda__435,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_mq(fun(B,A),fun(set(B),fun(B,$o)),Uu2),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu2,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_435
tff(fact_8615_ATP_Olambda__436,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_kt(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ~ dvd_dvd(B,aa(num,B,numeral_numeral(B),aa(num,num,bit0,one2)),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_436
tff(fact_8616_ATP_Olambda__437,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ad(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_437
tff(fact_8617_ATP_Olambda__438,axiom,
    ! [A: $tType,B: $tType,Uu2: list(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_adp(list(product_prod(A,B)),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,option(B),map_of(A,B,Uu2),Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_438
tff(fact_8618_ATP_Olambda__439,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_pf(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu2) ) ).

% ATP.lambda_439
tff(fact_8619_ATP_Olambda__440,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_gi(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu2) ) ).

% ATP.lambda_440
tff(fact_8620_ATP_Olambda__441,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_gp(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu2) ) ).

% ATP.lambda_441
tff(fact_8621_ATP_Olambda__442,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ok(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),divide_divide(A,Uub,Uu2)),Uua) ) ).

% ATP.lambda_442
tff(fact_8622_ATP_Olambda__443,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_oi(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),Uub)),Uua) ) ).

% ATP.lambda_443
tff(fact_8623_ATP_Olambda__444,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_oj(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Uub,Uu2)),Uua) ) ).

% ATP.lambda_444
tff(fact_8624_ATP_Olambda__445,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_oh(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),Uub)),Uua) ) ).

% ATP.lambda_445
tff(fact_8625_ATP_Olambda__446,axiom,
    ! [B: $tType,A: $tType,Uu2: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_eb(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu2),Uua),Uub)
    <=> 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),Uu2) ) ).

% ATP.lambda_446
tff(fact_8626_ATP_Olambda__447,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_abu(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> 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),Uu2) ) ).

% ATP.lambda_447
tff(fact_8627_ATP_Olambda__448,axiom,
    ! [A: $tType,Uu2: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aer(list(list(A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu2),Uub)),Uua) ).

% ATP.lambda_448
tff(fact_8628_ATP_Olambda__449,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_aq(nat,fun(complex,fun(complex,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu2) = Uua ) ) ).

% ATP.lambda_449
tff(fact_8629_ATP_Olambda__450,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_jg(complex,fun(nat,fun(complex,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_450
tff(fact_8630_ATP_Olambda__451,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu2: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_jk(A,fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu2),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_451
tff(fact_8631_ATP_Olambda__452,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dt(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_452
tff(fact_8632_ATP_Olambda__453,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ds(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_453
tff(fact_8633_ATP_Olambda__454,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_xs(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu2,Uub)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),aa(nat,int,Uua,Uub))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) ).

% ATP.lambda_454
tff(fact_8634_ATP_Olambda__455,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sb(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_455
tff(fact_8635_ATP_Olambda__456,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_dx(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_456
tff(fact_8636_ATP_Olambda__457,axiom,
    ! [Uu2: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gx(fun(nat,nat),fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),Uub)) ).

% ATP.lambda_457
tff(fact_8637_ATP_Olambda__458,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_wh(fun(nat,B),fun(B,fun(nat,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu2,Uub)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),Uub)) ) ).

% ATP.lambda_458
tff(fact_8638_ATP_Olambda__459,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_459
tff(fact_8639_ATP_Olambda__460,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ct(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_460
tff(fact_8640_ATP_Olambda__461,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cx(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_461
tff(fact_8641_ATP_Olambda__462,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: 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)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_462
tff(fact_8642_ATP_Olambda__463,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_463
tff(fact_8643_ATP_Olambda__464,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gn(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_464
tff(fact_8644_ATP_Olambda__465,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ao(fun(nat,$o),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,Uu2,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_465
tff(fact_8645_ATP_Olambda__466,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sq(fun(A,real),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_466
tff(fact_8646_ATP_Olambda__467,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_agi(fun(B,A),fun(B,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,Uu2,Uua)),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_467
tff(fact_8647_ATP_Olambda__468,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_yg(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_468
tff(fact_8648_ATP_Olambda__469,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_yb(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_469
tff(fact_8649_ATP_Olambda__470,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_yj(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_470
tff(fact_8650_ATP_Olambda__471,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fj(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = divide_divide(A,aa(B,A,Uu2,Uub),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_471
tff(fact_8651_ATP_Olambda__472,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = divide_divide(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_472
tff(fact_8652_ATP_Olambda__473,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = divide_divide(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_473
tff(fact_8653_ATP_Olambda__474,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aaq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = divide_divide(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_474
tff(fact_8654_ATP_Olambda__475,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_qx(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = divide_divide(A,aa(A,A,Uu2,Uub),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_475
tff(fact_8655_ATP_Olambda__476,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vz(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = divide_divide(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_476
tff(fact_8656_ATP_Olambda__477,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zh(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = divide_divide(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_477
tff(fact_8657_ATP_Olambda__478,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = divide_divide(B,aa(A,B,Uu2,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_478
tff(fact_8658_ATP_Olambda__479,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aba(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_479
tff(fact_8659_ATP_Olambda__480,axiom,
    ! [Uu2: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_kn(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_480
tff(fact_8660_ATP_Olambda__481,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aby(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_481
tff(fact_8661_ATP_Olambda__482,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_km(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_482
tff(fact_8662_ATP_Olambda__483,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fi(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_483
tff(fact_8663_ATP_Olambda__484,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_484
tff(fact_8664_ATP_Olambda__485,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aaw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_485
tff(fact_8665_ATP_Olambda__486,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aav(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_486
tff(fact_8666_ATP_Olambda__487,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rb(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_487
tff(fact_8667_ATP_Olambda__488,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vj(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_488
tff(fact_8668_ATP_Olambda__489,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vk(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_489
tff(fact_8669_ATP_Olambda__490,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zb(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_490
tff(fact_8670_ATP_Olambda__491,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vi(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_491
tff(fact_8671_ATP_Olambda__492,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_zg(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_492
tff(fact_8672_ATP_Olambda__493,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_493
tff(fact_8673_ATP_Olambda__494,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_494
tff(fact_8674_ATP_Olambda__495,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_mr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_495
tff(fact_8675_ATP_Olambda__496,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_zf(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu2,Uub)) ).

% ATP.lambda_496
tff(fact_8676_ATP_Olambda__497,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_497
tff(fact_8677_ATP_Olambda__498,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_wx(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu2,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_498
tff(fact_8678_ATP_Olambda__499,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_do(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_499
tff(fact_8679_ATP_Olambda__500,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bd(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_500
tff(fact_8680_ATP_Olambda__501,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ss(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_501
tff(fact_8681_ATP_Olambda__502,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aar(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_502
tff(fact_8682_ATP_Olambda__503,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_re(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_503
tff(fact_8683_ATP_Olambda__504,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wa(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_504
tff(fact_8684_ATP_Olambda__505,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_505
tff(fact_8685_ATP_Olambda__506,axiom,
    ! [B: $tType,A: $tType,Uu2: 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_ahx(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu2),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(A,set(B),Uu2,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_506
tff(fact_8686_ATP_Olambda__507,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ve(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_507
tff(fact_8687_ATP_Olambda__508,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vt(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_508
tff(fact_8688_ATP_Olambda__509,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ei(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uua,Uub)),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_509
tff(fact_8689_ATP_Olambda__510,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_cw(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu2,Uub)) ) ).

% ATP.lambda_510
tff(fact_8690_ATP_Olambda__511,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_bo(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu2,Uub)) ).

% ATP.lambda_511
tff(fact_8691_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_512
tff(fact_8692_ATP_Olambda__513,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_aay(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_513
tff(fact_8693_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_ux(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_514
tff(fact_8694_ATP_Olambda__515,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_uy(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_515
tff(fact_8695_ATP_Olambda__516,axiom,
    ! [B: $tType,A: $tType,Uu2: 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_ahy(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu2),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu2,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_516
tff(fact_8696_ATP_Olambda__517,axiom,
    ! [B: $tType,A: $tType,Uu2: 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_ahw(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu2),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu2,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_517
tff(fact_8697_ATP_Olambda__518,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_518
tff(fact_8698_ATP_Olambda__519,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bc(fun(B,A),fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu2,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_519
tff(fact_8699_ATP_Olambda__520,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_st(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_520
tff(fact_8700_ATP_Olambda__521,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aas(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_521
tff(fact_8701_ATP_Olambda__522,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rc(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_522
tff(fact_8702_ATP_Olambda__523,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_523
tff(fact_8703_ATP_Olambda__524,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: 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)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_524
tff(fact_8704_ATP_Olambda__525,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vf(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_525
tff(fact_8705_ATP_Olambda__526,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ka(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_526
tff(fact_8706_ATP_Olambda__527,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_rw(fun(real,real),fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = powr(real,aa(real,real,Uu2,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_527
tff(fact_8707_ATP_Olambda__528,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_tv(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_528
tff(fact_8708_ATP_Olambda__529,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_abc(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,log(aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_529
tff(fact_8709_ATP_Olambda__530,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_xd(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,log(aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_530
tff(fact_8710_ATP_Olambda__531,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vw(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,log(aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_531
tff(fact_8711_ATP_Olambda__532,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: 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_ail(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu2),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu2,Uub)),aa(C,B,Uua,Uub)) ).

% ATP.lambda_532
tff(fact_8712_ATP_Olambda__533,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: 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_aaz(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu2),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_533
tff(fact_8713_ATP_Olambda__534,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: 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_uk(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu2),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_534
tff(fact_8714_ATP_Olambda__535,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: 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_ur(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu2),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_535
tff(fact_8715_ATP_Olambda__536,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aev(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & aa(A,$o,Uu2,Uub) ) ) ).

% ATP.lambda_536
tff(fact_8716_ATP_Olambda__537,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aez(fun(A,B),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,B,Uu2,Uua) = aa(A,B,Uu2,Uub) ) ) ).

% ATP.lambda_537
tff(fact_8717_ATP_Olambda__538,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_age(A,fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu2) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_538
tff(fact_8718_ATP_Olambda__539,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_nj(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_539
tff(fact_8719_ATP_Olambda__540,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yx(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_540
tff(fact_8720_ATP_Olambda__541,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_ro(fun(A,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu2,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_541
tff(fact_8721_ATP_Olambda__542,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu2: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_ut(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu2),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)),Uu2) ) ).

% ATP.lambda_542
tff(fact_8722_ATP_Olambda__543,axiom,
    ! [Uu2: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ap(fun(nat,nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu2,Uub)),Uua) ) ).

% ATP.lambda_543
tff(fact_8723_ATP_Olambda__544,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yh(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_544
tff(fact_8724_ATP_Olambda__545,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aac(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_545
tff(fact_8725_ATP_Olambda__546,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yp(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_546
tff(fact_8726_ATP_Olambda__547,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aaa(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_547
tff(fact_8727_ATP_Olambda__548,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_bf(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu2,Uub),Uua) ) ).

% ATP.lambda_548
tff(fact_8728_ATP_Olambda__549,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_di(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = divide_divide(A,aa(nat,A,Uu2,Uub),Uua) ) ).

% ATP.lambda_549
tff(fact_8729_ATP_Olambda__550,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_be(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = divide_divide(A,aa(B,A,Uu2,Uub),Uua) ) ).

% ATP.lambda_550
tff(fact_8730_ATP_Olambda__551,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qy(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = divide_divide(A,aa(A,A,Uu2,Uub),Uua) ) ).

% ATP.lambda_551
tff(fact_8731_ATP_Olambda__552,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_nq(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = divide_divide(B,aa(A,B,Uu2,Uub),Uua) ) ).

% ATP.lambda_552
tff(fact_8732_ATP_Olambda__553,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_va(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = divide_divide(B,aa(A,B,Uu2,Uub),Uua) ) ).

% ATP.lambda_553
tff(fact_8733_ATP_Olambda__554,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wn(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = divide_divide(A,aa(nat,A,Uua,Uub),Uu2) ) ).

% ATP.lambda_554
tff(fact_8734_ATP_Olambda__555,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_yn(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_555
tff(fact_8735_ATP_Olambda__556,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yl(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_556
tff(fact_8736_ATP_Olambda__557,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aab(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_557
tff(fact_8737_ATP_Olambda__558,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dm(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_558
tff(fact_8738_ATP_Olambda__559,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ba(fun(B,A),fun(A,fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_559
tff(fact_8739_ATP_Olambda__560,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_sv(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_560
tff(fact_8740_ATP_Olambda__561,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aal(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_561
tff(fact_8741_ATP_Olambda__562,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aat(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_562
tff(fact_8742_ATP_Olambda__563,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ra(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_563
tff(fact_8743_ATP_Olambda__564,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vm(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_564
tff(fact_8744_ATP_Olambda__565,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vh(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_565
tff(fact_8745_ATP_Olambda__566,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vp(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_566
tff(fact_8746_ATP_Olambda__567,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ti(real,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu2) ) ).

% ATP.lambda_567
tff(fact_8747_ATP_Olambda__568,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ed(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_568
tff(fact_8748_ATP_Olambda__569,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wm(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_569
tff(fact_8749_ATP_Olambda__570,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uh(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_570
tff(fact_8750_ATP_Olambda__571,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cy(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_571
tff(fact_8751_ATP_Olambda__572,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_pr(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(B,set(A),Uu2,Uub)),Uua) ).

% ATP.lambda_572
tff(fact_8752_ATP_Olambda__573,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vc(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_573
tff(fact_8753_ATP_Olambda__574,axiom,
    ! [Uu2: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_rs(fun(real,real),fun(nat,fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uu2,Uub)),Uua) ).

% ATP.lambda_574
tff(fact_8754_ATP_Olambda__575,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: fun(B,A),Uua: nat,Uub: B] : aa(B,A,aa(nat,fun(B,A),aTP_Lamp_fk(fun(B,A),fun(nat,fun(B,A)),Uu2),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(B,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_575
tff(fact_8755_ATP_Olambda__576,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_tl(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_576
tff(fact_8756_ATP_Olambda__577,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_aax(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_577
tff(fact_8757_ATP_Olambda__578,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_rp(fun(A,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_578
tff(fact_8758_ATP_Olambda__579,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_uv(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_579
tff(fact_8759_ATP_Olambda__580,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_zi(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_580
tff(fact_8760_ATP_Olambda__581,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_uz(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_581
tff(fact_8761_ATP_Olambda__582,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_uw(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_582
tff(fact_8762_ATP_Olambda__583,axiom,
    ! [Uu2: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_zy(nat,fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uua,Uub)),Uu2) ).

% ATP.lambda_583
tff(fact_8763_ATP_Olambda__584,axiom,
    ! [A: $tType,Uu2: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uj(nat,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uua,Uub)),Uu2) ).

% ATP.lambda_584
tff(fact_8764_ATP_Olambda__585,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_abx(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_585
tff(fact_8765_ATP_Olambda__586,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_pa(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_586
tff(fact_8766_ATP_Olambda__587,axiom,
    ! [Uu2: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_rv(fun(real,real),fun(real,fun(real,real)),Uu2),Uua),Uub) = powr(real,aa(real,real,Uu2,Uub),Uua) ).

% ATP.lambda_587
tff(fact_8767_ATP_Olambda__588,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(B,A),Uua: int,Uub: B] : aa(B,A,aa(int,fun(B,A),aTP_Lamp_agv(fun(B,A),fun(int,fun(B,A)),Uu2),Uua),Uub) = power_int(A,aa(B,A,Uu2,Uub),Uua) ) ).

% ATP.lambda_588
tff(fact_8768_ATP_Olambda__589,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_agz(fun(A,A),fun(int,fun(A,A)),Uu2),Uua),Uub) = power_int(A,aa(A,A,Uu2,Uub),Uua) ) ).

% ATP.lambda_589
tff(fact_8769_ATP_Olambda__590,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ki(nat,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = 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),Uu2),Uub))) ) ).

% ATP.lambda_590
tff(fact_8770_ATP_Olambda__591,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_dy(real,fun(fun(nat,A),fun(nat,real)),Uu2),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),Uu2),Uub)) ) ).

% ATP.lambda_591
tff(fact_8771_ATP_Olambda__592,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu2: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_nk(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu2,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_592
tff(fact_8772_ATP_Olambda__593,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ik(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = 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),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_593
tff(fact_8773_ATP_Olambda__594,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yw(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_594
tff(fact_8774_ATP_Olambda__595,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: fun(list(A),fun(list(A),$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_acj(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu2),Uua),Uub)
        <=> ( ? [Y5: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),cons(A,Y5),Ys4) ) )
            | ? [X3: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),cons(A,X3),Xs3) )
                & ( Uub = aa(list(A),list(A),cons(A,Y5),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y5) )
            | ? [X3: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),cons(A,X3),Xs3) )
                & ( Uub = aa(list(A),list(A),cons(A,Y5),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y5)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X3)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uu2,Xs3),Ys4) ) ) ) ) ).

% ATP.lambda_595
tff(fact_8775_ATP_Olambda__596,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kg(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),Uu2),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_596
tff(fact_8776_ATP_Olambda__597,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kh(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),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_597
tff(fact_8777_ATP_Olambda__598,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_kf(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),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_598
tff(fact_8778_ATP_Olambda__599,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_agy(A,fun(int,fun(A,A)),Uu2),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),aa(int,A,ring_1_of_int(A),Uua)),power_int(A,Uu2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Uua),one_one(int))))) ) ).

% ATP.lambda_599
tff(fact_8779_ATP_Olambda__600,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ci(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_600
tff(fact_8780_ATP_Olambda__601,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ch(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_601
tff(fact_8781_ATP_Olambda__602,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_de(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_602
tff(fact_8782_ATP_Olambda__603,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aea(fun(A,option(B)),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),dom(A,B,Uu2))) ) ).

% ATP.lambda_603
tff(fact_8783_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aeb(fun(A,option(B)),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),dom(A,B,Uu2))) ) ).

% ATP.lambda_604
tff(fact_8784_ATP_Olambda__605,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_xh(real,fun(real,fun(nat,real)),Uu2),Uua),Uub) = divide_divide(real,Uua,aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uub)) ).

% ATP.lambda_605
tff(fact_8785_ATP_Olambda__606,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_br(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_606
tff(fact_8786_ATP_Olambda__607,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_cd(nat,fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_607
tff(fact_8787_ATP_Olambda__608,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: A,Uua: B,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(B,fun(C,product_prod(A,product_prod(B,C))),aa(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_adl(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu2),Uua),Uub) = 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)),Uu2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).

% ATP.lambda_608
tff(fact_8788_ATP_Olambda__609,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: B,Uua: A,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(A,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_adm(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu2),Uua),Uub) = 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)),Uua),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu2),Uub)) ).

% ATP.lambda_609
tff(fact_8789_ATP_Olambda__610,axiom,
    ! [A: $tType,Uu2: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_pj(A,fun(list(A),fun(nat,list(A))),Uu2),Uua),Uub) = aa(list(A),list(A),cons(A,Uu2),take(A,Uub,Uua)) ).

% ATP.lambda_610
tff(fact_8790_ATP_Olambda__611,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,option(A)),Uua: list(B),Uub: A] : aa(A,list(A),aa(list(B),fun(A,list(A)),aTP_Lamp_afp(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu2),Uua),Uub) = aa(list(A),list(A),cons(A,Uub),map_filter(B,A,Uu2,Uua)) ).

% ATP.lambda_611
tff(fact_8791_ATP_Olambda__612,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_yq(B,fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu2),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_612
tff(fact_8792_ATP_Olambda__613,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yi(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_613
tff(fact_8793_ATP_Olambda__614,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yf(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_614
tff(fact_8794_ATP_Olambda__615,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yc(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_615
tff(fact_8795_ATP_Olambda__616,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_lz(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = divide_divide(A,Uua,aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_616
tff(fact_8796_ATP_Olambda__617,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ym(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_617
tff(fact_8797_ATP_Olambda__618,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_agn(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_618
tff(fact_8798_ATP_Olambda__619,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ya(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_619
tff(fact_8799_ATP_Olambda__620,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_yk(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_620
tff(fact_8800_ATP_Olambda__621,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aak(B,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu2),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_621
tff(fact_8801_ATP_Olambda__622,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_dj(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_622
tff(fact_8802_ATP_Olambda__623,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bb(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_623
tff(fact_8803_ATP_Olambda__624,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ee(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_624
tff(fact_8804_ATP_Olambda__625,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wl(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_625
tff(fact_8805_ATP_Olambda__626,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ui(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_626
tff(fact_8806_ATP_Olambda__627,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dl(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_627
tff(fact_8807_ATP_Olambda__628,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_su(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_628
tff(fact_8808_ATP_Olambda__629,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aau(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_629
tff(fact_8809_ATP_Olambda__630,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qz(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu2,Uub)) ) ).

% ATP.lambda_630
tff(fact_8810_ATP_Olambda__631,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vl(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_631
tff(fact_8811_ATP_Olambda__632,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vg(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_632
tff(fact_8812_ATP_Olambda__633,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vo(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_633
tff(fact_8813_ATP_Olambda__634,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aae(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_634
tff(fact_8814_ATP_Olambda__635,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_px(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu2),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_635
tff(fact_8815_ATP_Olambda__636,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_ft(A,fun(fun(B,nat),fun(B,A)),Uu2),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_636
tff(fact_8816_ATP_Olambda__637,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu2: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xm(fun(A,nat),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),aa(A,nat,Uu2,Uub)) ) ).

% ATP.lambda_637
tff(fact_8817_ATP_Olambda__638,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vs(B,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu2),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_638
tff(fact_8818_ATP_Olambda__639,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_nr(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> dvd_dvd(B,Uua,aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_639
tff(fact_8819_ATP_Olambda__640,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_adf(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu2),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu2,Uub)) ).

% ATP.lambda_640
tff(fact_8820_ATP_Olambda__641,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_agg(fun(list(A),A),fun(list(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu2,Uua) ) ) ) ).

% ATP.lambda_641
tff(fact_8821_ATP_Olambda__642,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_go(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_642
tff(fact_8822_ATP_Olambda__643,axiom,
    ! [A: $tType,Uu2: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_afe(list(A),fun(set(nat),fun(A,$o)),Uu2),Uua),Uub)
    <=> member(A,Uub,aa(list(A),set(A),set2(A),nths(A,Uu2,Uua))) ) ).

% ATP.lambda_643
tff(fact_8823_ATP_Olambda__644,axiom,
    ! [A: $tType,B: $tType,Uu2: set(B),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_ahv(set(B),fun(set(B),fun(A,set(B))),Uu2),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uu2),Uua) ).

% ATP.lambda_644
tff(fact_8824_ATP_Olambda__645,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_afn(list(A),fun(A,fun(list(A),list(A))),Uu2),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uub),Uu2) ).

% ATP.lambda_645
tff(fact_8825_ATP_Olambda__646,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_ahl(B,fun(set(B),fun(A,set(B))),Uu2),Uua),Uub) = aa(set(B),set(B),insert(B,Uu2),Uua) ).

% ATP.lambda_646
tff(fact_8826_ATP_Olambda__647,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu2: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_ahr(fun(D,B),fun(set(D),fun(A,set(B))),Uu2),Uua),Uub) = image(D,B,Uu2,Uua) ).

% ATP.lambda_647
tff(fact_8827_ATP_Olambda__648,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fx(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_648
tff(fact_8828_ATP_Olambda__649,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_649
tff(fact_8829_ATP_Olambda__650,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wu(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_650
tff(fact_8830_ATP_Olambda__651,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wq(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uua)) ) ).

% ATP.lambda_651
tff(fact_8831_ATP_Olambda__652,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_un(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_652
tff(fact_8832_ATP_Olambda__653,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_qv(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_653
tff(fact_8833_ATP_Olambda__654,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_xz(fun(nat,$o),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_654
tff(fact_8834_ATP_Olambda__655,axiom,
    ! [A: $tType,Uu2: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_pk(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu2),Uua),Uub) = aa(nat,set(A),Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).

% ATP.lambda_655
tff(fact_8835_ATP_Olambda__656,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dk(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_656
tff(fact_8836_ATP_Olambda__657,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wp(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_657
tff(fact_8837_ATP_Olambda__658,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_abz(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_658
tff(fact_8838_ATP_Olambda__659,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fu(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_659
tff(fact_8839_ATP_Olambda__660,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bq(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_660
tff(fact_8840_ATP_Olambda__661,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_yo(fun(A,$o),fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_661
tff(fact_8841_ATP_Olambda__662,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_um(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_662
tff(fact_8842_ATP_Olambda__663,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_wd(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_663
tff(fact_8843_ATP_Olambda__664,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_rh(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_664
tff(fact_8844_ATP_Olambda__665,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu2: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_bn(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(product_prod(A,B),C,Uu2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_665
tff(fact_8845_ATP_Olambda__666,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_uq(B,fun(fun(B,C),fun(B,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu2),Uub)) ) ).

% ATP.lambda_666
tff(fact_8846_ATP_Olambda__667,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_uo(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),Uub)) ) ).

% ATP.lambda_667
tff(fact_8847_ATP_Olambda__668,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_eg(nat,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu2)) ) ).

% ATP.lambda_668
tff(fact_8848_ATP_Olambda__669,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_th(fun(real,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,Uu2,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_669
tff(fact_8849_ATP_Olambda__670,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,Uu2: 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_lr(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),Uu2),Uua),Uub) = aa(D,fun(B,C),Uu2,aa(A,D,Uua,Uub)) ).

% ATP.lambda_670
tff(fact_8850_ATP_Olambda__671,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu2: fun(C,B),Uua: fun(A,C),Uub: A] : aa(A,B,aa(fun(A,C),fun(A,B),aTP_Lamp_nw(fun(C,B),fun(fun(A,C),fun(A,B)),Uu2),Uua),Uub) = aa(C,B,Uu2,aa(A,C,Uua,Uub)) ).

% ATP.lambda_671
tff(fact_8851_ATP_Olambda__672,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(num,B),Uub: num] : aa(num,A,aa(fun(num,B),fun(num,A),aTP_Lamp_nl(fun(B,A),fun(fun(num,B),fun(num,A)),Uu2),Uua),Uub) = aa(B,A,Uu2,aa(num,B,Uua,Uub)) ).

% ATP.lambda_672
tff(fact_8852_ATP_Olambda__673,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_lh(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu2),Uua),Uub) = aa(B,A,Uu2,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_673
tff(fact_8853_ATP_Olambda__674,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_mw(fun(B,A),fun(fun(C,B),fun(C,A)),Uu2),Uua),Uub) = aa(B,A,Uu2,aa(C,B,Uua,Uub)) ).

% ATP.lambda_674
tff(fact_8854_ATP_Olambda__675,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu2: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_aag(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,Uu2,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_675
tff(fact_8855_ATP_Olambda__676,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aah(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_676
tff(fact_8856_ATP_Olambda__677,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rl(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uu2,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_677
tff(fact_8857_ATP_Olambda__678,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_acb(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_678
tff(fact_8858_ATP_Olambda__679,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_up(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_679
tff(fact_8859_ATP_Olambda__680,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_rk(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu2,Uub)) ) ).

% ATP.lambda_680
tff(fact_8860_ATP_Olambda__681,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_jd(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_681
tff(fact_8861_ATP_Olambda__682,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_jc(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_682
tff(fact_8862_ATP_Olambda__683,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ol(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_683
tff(fact_8863_ATP_Olambda__684,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu2: fun(D,set(B)),Uua: D,Uub: A] : aa(A,set(B),aa(D,fun(A,set(B)),aTP_Lamp_ahm(fun(D,set(B)),fun(D,fun(A,set(B))),Uu2),Uua),Uub) = aa(D,set(B),Uu2,Uua) ).

% ATP.lambda_684
tff(fact_8864_ATP_Olambda__685,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_wj(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu2),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_wi(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu2),Uua),Uub)) ) ).

% ATP.lambda_685
tff(fact_8865_ATP_Olambda__686,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ug(fun(nat,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_uf(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub)) ) ).

% ATP.lambda_686
tff(fact_8866_ATP_Olambda__687,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: A,Uua: set(A),Uub: A] : aa(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aaj(A,fun(set(A),fun(A,filter(A))),Uu2),Uua),Uub) = 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)),set_ord_greaterThan(A,Uub)),Uua)),aa(set(A),set(A),insert(A,Uu2),bot_bot(set(A))))) ) ).

% ATP.lambda_687
tff(fact_8867_ATP_Olambda__688,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: A,Uua: set(A),Uub: A] : aa(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_aai(A,fun(set(A),fun(A,filter(A))),Uu2),Uua),Uub) = 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)),set_ord_lessThan(A,Uub)),Uua)),aa(set(A),set(A),insert(A,Uu2),bot_bot(set(A))))) ) ).

% ATP.lambda_688
tff(fact_8868_ATP_Olambda__689,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A,Uua: set(A),Uub: set(A)] : aa(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_aao(A,fun(set(A),fun(set(A),filter(A))),Uu2),Uua),Uub) = 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,Uu2),bot_bot(set(A))))) ) ).

% ATP.lambda_689
tff(fact_8869_ATP_Olambda__690,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_cu(fun(nat,A),fun(A,fun(nat,real)),Uu2),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub))) ) ).

% ATP.lambda_690
tff(fact_8870_ATP_Olambda__691,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_gf(fun(A,B),fun(fun(A,B),fun(A,real)),Uu2),Uua),Uub) = real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_691
tff(fact_8871_ATP_Olambda__692,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_xr(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = cos(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu2,Uub)),Uua)) ).

% ATP.lambda_692
tff(fact_8872_ATP_Olambda__693,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ns(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ~ dvd_dvd(B,Uua,aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_693
tff(fact_8873_ATP_Olambda__694,axiom,
    ! [A: $tType,B: $tType,D: $tType,Uu2: fun(D,set(B)),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_ahp(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu2),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),image(D,set(B),Uu2,Uua)) ).

% ATP.lambda_694
tff(fact_8874_ATP_Olambda__695,axiom,
    ! [B: $tType,A: $tType,Uu2: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_lc(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu2),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_lb(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uua),Uub))) ).

% ATP.lambda_695
tff(fact_8875_ATP_Olambda__696,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu2: fun(B,real),Uua: fun(real,A),Uub: B] : aa(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_ud(fun(B,real),fun(fun(real,A),fun(B,real)),Uu2),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uua,aa(B,real,Uu2,Uub)))) ) ).

% ATP.lambda_696
tff(fact_8876_ATP_Olambda__697,axiom,
    ! [A: $tType,B: $tType,Uu2: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_pb(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu2),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu2),I3)),aa(nat,B,nth(B,Uua),I3)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu2)),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).

% ATP.lambda_697
tff(fact_8877_ATP_Olambda__698,axiom,
    ! [A: $tType,Uu2: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_ni(list(A),fun(set(nat),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu2),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu2))
          & member(nat,I3,Uua) ) ) ).

% ATP.lambda_698
tff(fact_8878_ATP_Olambda__699,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu2: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_act(set(A),fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> ? [A6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A6) )
              & member(A,A6,Uu2) ) ) ) ).

% ATP.lambda_699
tff(fact_8879_ATP_Olambda__700,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu2: A,Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_pu(A,fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ? [B6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu2),B6) )
              & member(A,B6,Uua) ) ) ) ).

% ATP.lambda_700
tff(fact_8880_ATP_Olambda__701,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu2: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ada(set(A),fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> ? [A6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A6) )
              & member(A,A6,Uu2) ) ) ) ).

% ATP.lambda_701
tff(fact_8881_ATP_Olambda__702,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_od(fun(B,A),fun(set(B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [X3: B] :
          ( ( Uub = aa(B,A,Uu2,X3) )
          & member(B,X3,Uua) ) ) ).

% ATP.lambda_702
tff(fact_8882_ATP_Olambda__703,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: fun(B,$o),Uub: set(A)] :
      ( aa(set(A),$o,aa(fun(B,$o),fun(set(A),$o),aTP_Lamp_pp(fun(B,set(A)),fun(fun(B,$o),fun(set(A),$o)),Uu2),Uua),Uub)
    <=> ? [X3: B] :
          ( ( Uub = aa(B,set(A),Uu2,X3) )
          & aa(B,$o,Uua,X3) ) ) ).

% ATP.lambda_703
tff(fact_8883_ATP_Olambda__704,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_oe(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [X3: B] :
          ( ( Uub = aa(B,A,Uu2,X3) )
          & aa(B,$o,Uua,X3) ) ) ).

% ATP.lambda_704
tff(fact_8884_ATP_Olambda__705,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,$o),Uua: fun(A,B),Uub: B] :
      ( aa(B,$o,aa(fun(A,B),fun(B,$o),aTP_Lamp_am(fun(A,$o),fun(fun(A,B),fun(B,$o)),Uu2),Uua),Uub)
    <=> ? [X3: A] :
          ( ( Uub = aa(A,B,Uua,X3) )
          & aa(A,$o,Uu2,X3) ) ) ).

% ATP.lambda_705
tff(fact_8885_ATP_Olambda__706,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_yy(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),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,Uu2),set_or7035219750837199246ssThan(nat,Uub,N4)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_706
tff(fact_8886_ATP_Olambda__707,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_xx(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),Uua),Uub)
        <=> ! [A6: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),A6)
             => ! [B6: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A6),B6)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu2),set_or3652927894154168847AtMost(nat,A6,B6)))),aa(nat,real,Uua,A6)) ) ) ) ) ).

% ATP.lambda_707
tff(fact_8887_ATP_Olambda__708,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,$o),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aj(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu2),Uua),Uub)
    <=> ? [Y5: A] :
          ( aa(A,$o,Uu2,Y5)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),Y5) ) ) ).

% ATP.lambda_708
tff(fact_8888_ATP_Olambda__709,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_pq(fun(B,set(A)),fun(fun(B,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [X3: B] :
          ( aa(B,$o,Uua,X3)
          & member(A,Uub,aa(B,set(A),Uu2,X3)) ) ) ).

% ATP.lambda_709
tff(fact_8889_ATP_Olambda__710,axiom,
    ! [A: $tType,Uu2: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_acm(fun(A,A),fun(A,fun(A,$o)),Uu2),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),Uu2),Uua) ) ).

% ATP.lambda_710
tff(fact_8890_ATP_Olambda__711,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_qg(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ? [A6: A,V5: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),cons(A,A6),V5)) )
          | ? [U4: list(A),Aa3: A,B6: A,Va4: list(A),W3: list(A)] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa3),B6),Uu2)
              & ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U4),aa(list(A),list(A),cons(A,Aa3),Va4)) )
              & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U4),aa(list(A),list(A),cons(A,B6),W3)) ) ) ) ) ).

% ATP.lambda_711
tff(fact_8891_ATP_Olambda__712,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu2: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_acs(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ? [A6: A,B6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A6),B6) )
              & member(A,A6,Uu2)
              & member(A,B6,Uua) ) ) ) ).

% ATP.lambda_712
tff(fact_8892_ATP_Olambda__713,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu2: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_acz(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ? [A6: A,B6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A6),B6) )
              & member(A,A6,Uu2)
              & member(A,B6,Uua) ) ) ) ).

% ATP.lambda_713
tff(fact_8893_ATP_Olambda__714,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(list(A)),fun(list(A),$o),aTP_Lamp_po(set(A),fun(set(list(A)),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ? [X3: A,Xs3: list(A)] :
          ( ( Uub = aa(list(A),list(A),cons(A,X3),Xs3) )
          & member(A,X3,Uu2)
          & member(list(A),Xs3,Uua) ) ) ).

% ATP.lambda_714
tff(fact_8894_ATP_Olambda__715,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_qf(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ? [Us2: list(A),Z2: A,Z7: A,Vs3: list(A)] :
          ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),cons(A,Z2),Vs3)) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Z7),Uu2)
          & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),cons(A,Z7),Vs3)) ) ) ) ).

% ATP.lambda_715
tff(fact_8895_ATP_Olambda__716,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_it(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub)
            & 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),real_V8093663219630862766scaleR(A,divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),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),Uu2),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_716
tff(fact_8896_ATP_Olambda__717,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ic(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            ( dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub)
            & ~ 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),real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),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),Uu2),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_717
tff(fact_8897_ATP_Olambda__718,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ie(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(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),real_V8093663219630862766scaleR(A,divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),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),Uu2),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_718
tff(fact_8898_ATP_Olambda__719,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: 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_aeh(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu2),Uua),Uub),Uuc) = $ite(aa(product_prod(C,A),A,product_snd(C,A),Uu2) = 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),Uu2)),Uub)),Uuc),Uuc) ).

% ATP.lambda_719
tff(fact_8899_ATP_Olambda__720,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_hb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu2,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_720
tff(fact_8900_ATP_Olambda__721,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_fz(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu2,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_721
tff(fact_8901_ATP_Olambda__722,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: 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_aaf(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uuc),Uu2),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_722
tff(fact_8902_ATP_Olambda__723,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_ga(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_723
tff(fact_8903_ATP_Olambda__724,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_hc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_724
tff(fact_8904_ATP_Olambda__725,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: 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_ej(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(member(nat,Uuc,Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu2,Uuc)) ) ).

% ATP.lambda_725
tff(fact_8905_ATP_Olambda__726,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: 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_ji(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_726
tff(fact_8906_ATP_Olambda__727,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: 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_jh(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_727
tff(fact_8907_ATP_Olambda__728,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: 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_le(A,fun(fun(A,B),fun(B,fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_728
tff(fact_8908_ATP_Olambda__729,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A,Uub: list(A),Uuc: list(A)] :
      aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_agu(fun(A,$o),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu2,Uua),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)),aa(list(A),list(A),cons(A,Uua),Uub)),Uuc),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)),Uub),aa(list(A),list(A),cons(A,Uua),Uuc))) ).

% ATP.lambda_729
tff(fact_8909_ATP_Olambda__730,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,$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_np(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu2,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_730
tff(fact_8910_ATP_Olambda__731,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,$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_nn(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu2,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_731
tff(fact_8911_ATP_Olambda__732,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: 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_ael(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu2),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_aek(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu2) ).

% ATP.lambda_732
tff(fact_8912_ATP_Olambda__733,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: 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_aeg(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu2),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_aef(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu2) ).

% ATP.lambda_733
tff(fact_8913_ATP_Olambda__734,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu2: 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_lv(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu2),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,Uu2,aa(nat,B,nth(B,Uub),Uuc))) ) ).

% ATP.lambda_734
tff(fact_8914_ATP_Olambda__735,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(A,$o)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_agh(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> aa(A,$o,aa(A,fun(A,$o),Uu2,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ).

% ATP.lambda_735
tff(fact_8915_ATP_Olambda__736,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,Uu2: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(D,C),Uuc: D] : aa(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_adj(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),Uu2),Uua),Uub),Uuc) = aa(C,A,aa(B,fun(C,A),Uu2,aa(D,B,Uua,Uuc)),aa(D,C,Uub,Uuc)) ).

% ATP.lambda_736
tff(fact_8916_ATP_Olambda__737,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(A,fun(B,A)),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,A,aa(A,fun(C,A),aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_afy(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(B,A,aa(A,fun(B,A),Uu2,Uub),aa(C,B,Uua,Uuc)) ).

% ATP.lambda_737
tff(fact_8917_ATP_Olambda__738,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cq(A,fun(A,fun(nat,fun(nat,A))),Uu2),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_cp(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),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_738
tff(fact_8918_ATP_Olambda__739,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu2: 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_az(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu2),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_ay(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uuc)),Uub) ) ).

% ATP.lambda_739
tff(fact_8919_ATP_Olambda__740,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: nat,Uua: A,Uub: A,Uuc: nat] :
          aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_hf(nat,fun(A,fun(A,fun(nat,A))),Uu2),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 = Uu2,one_one(A),zero_zero(A)) )),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_740
tff(fact_8920_ATP_Olambda__741,axiom,
    ! [Uu2: 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_gz(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),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_gy(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_741
tff(fact_8921_ATP_Olambda__742,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_gv(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu2),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_gu(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_742
tff(fact_8922_ATP_Olambda__743,axiom,
    ! [Uu2: 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_sj(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu2,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_743
tff(fact_8923_ATP_Olambda__744,axiom,
    ! [Uu2: 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_sh(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu2,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_744
tff(fact_8924_ATP_Olambda__745,axiom,
    ! [Uu2: 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_sf(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu2,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_745
tff(fact_8925_ATP_Olambda__746,axiom,
    ! [Uu2: 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_sg(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu2,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_746
tff(fact_8926_ATP_Olambda__747,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_da(A,fun(A,fun(nat,fun(nat,A))),Uu2),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),Uu2),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),Uu2),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uub)) ) ).

% ATP.lambda_747
tff(fact_8927_ATP_Olambda__748,axiom,
    ! [Uu2: 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_md(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu2),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),Uu2),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),Uu2),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).

% ATP.lambda_748
tff(fact_8928_ATP_Olambda__749,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_cp(A,fun(A,fun(nat,fun(nat,A))),Uu2),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),Uu2)),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_749
tff(fact_8929_ATP_Olambda__750,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: 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_xo(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = divide_divide(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu2,Uua)),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_750
tff(fact_8930_ATP_Olambda__751,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_qk(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu2,Uub)),aa(A,nat,Uu2,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu2,Uub)),aa(A,nat,Uu2,Uuc))
          & 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_751
tff(fact_8931_ATP_Olambda__752,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hx(A,fun(A,fun(nat,fun(nat,A))),Uu2),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,Uu2,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_752
tff(fact_8932_ATP_Olambda__753,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hv(nat,fun(nat,fun(nat,fun(nat,nat))),Uu2),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),Uu2),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_753
tff(fact_8933_ATP_Olambda__754,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_hw(A,fun(A,fun(nat,fun(nat,A))),Uu2),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),Uu2),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_754
tff(fact_8934_ATP_Olambda__755,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ii(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(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),Uu2),Uuc))),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_755
tff(fact_8935_ATP_Olambda__756,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_qj(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X3: A,Y5: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),cons(A,X3),Xs5)) )
            & ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),cons(A,Y5),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y5),Uu2) ) ) ) ).

% ATP.lambda_756
tff(fact_8936_ATP_Olambda__757,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dc(A,fun(nat,fun(A,fun(nat,A))),Uu2),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),Uu2),Uuc)) ) ).

% ATP.lambda_757
tff(fact_8937_ATP_Olambda__758,axiom,
    ! [A: $tType,B: $tType,Uu2: set(B),Uua: fun(A,fun(B,$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_lb(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_758
tff(fact_8938_ATP_Olambda__759,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(A,fun(B,$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_la(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_759
tff(fact_8939_ATP_Olambda__760,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_db(A,fun(A,fun(nat,fun(nat,A))),Uu2),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),Uu2),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),Uu2),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),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc)))) ) ).

% ATP.lambda_760
tff(fact_8940_ATP_Olambda__761,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_om(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_761
tff(fact_8941_ATP_Olambda__762,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_dd(A,fun(nat,fun(A,fun(nat,A))),Uu2),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),Uu2),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_762
tff(fact_8942_ATP_Olambda__763,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_hs(nat,fun(nat,fun(nat,fun(nat,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu2),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_763
tff(fact_8943_ATP_Olambda__764,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_jo(int,fun(int,fun(int,fun(int,$o))),Uu2),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),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_764
tff(fact_8944_ATP_Olambda__765,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_mj(nat,fun(nat,fun(nat,fun(nat,$o))),Uu2),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),Uu2),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_765
tff(fact_8945_ATP_Olambda__766,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_jm(int,fun(int,fun(int,fun(int,$o))),Uu2),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),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_766
tff(fact_8946_ATP_Olambda__767,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_mh(nat,fun(nat,fun(nat,fun(nat,$o))),Uu2),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),Uu2),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_767
tff(fact_8947_ATP_Olambda__768,axiom,
    ! [Uu2: 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_ml(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu2),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),Uu2),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ).

% ATP.lambda_768
tff(fact_8948_ATP_Olambda__769,axiom,
    ! [Uu2: 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_mn(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu2),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),Uu2),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).

% ATP.lambda_769
tff(fact_8949_ATP_Olambda__770,axiom,
    ! [A: $tType,B: $tType,Uu2: A,Uua: list(A),Uub: B,Uuc: list(B)] : aa(list(B),list(product_prod(A,B)),aa(B,fun(list(B),list(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_abp(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu2),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu2),Uub)),zip(A,B,Uua,Uuc)) ).

% ATP.lambda_770
tff(fact_8950_ATP_Olambda__771,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: list(B),Uub: A,Uuc: list(A)] : aa(list(A),list(product_prod(A,B)),aa(A,fun(list(A),list(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_abm(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu2),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),cons(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uu2)),zip(A,B,Uuc,Uua)) ).

% ATP.lambda_771
tff(fact_8951_ATP_Olambda__772,axiom,
    ! [A: $tType,B: $tType,Uu2: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_jb(A,fun(B,fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ( Uu2 = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_772
tff(fact_8952_ATP_Olambda__773,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_pn(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> 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)),Uu2,Uua)) ) ).

% ATP.lambda_773
tff(fact_8953_ATP_Olambda__774,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_as(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu2)
            & ( 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_774
tff(fact_8954_ATP_Olambda__775,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_au(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu2)
            & ( 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_775
tff(fact_8955_ATP_Olambda__776,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_za(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu2),Uua),Uub),Uuc) = real_V8093663219630862766scaleR(B,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,Uu2,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))) ) ).

% ATP.lambda_776
tff(fact_8956_ATP_Olambda__777,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: 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_du(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,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_777
tff(fact_8957_ATP_Olambda__778,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: 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_kl(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_778
tff(fact_8958_ATP_Olambda__779,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,$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,$o)),aTP_Lamp_ahs(fun(A,$o),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu2,Uub)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_779
tff(fact_8959_ATP_Olambda__780,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,Uu2: 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_ahn(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),Uu2),Uua),Uub),Uuc) = product_Sigma(A,B,aa(C,set(A),Uu2,Uub),aa(D,fun(A,set(B)),aTP_Lamp_ahm(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).

% ATP.lambda_780
tff(fact_8960_ATP_Olambda__781,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: 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_uf(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(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_781
tff(fact_8961_ATP_Olambda__782,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu2: 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_wi(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu2),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,Uu2,Uub)),Uuc)) ) ).

% ATP.lambda_782
tff(fact_8962_ATP_Olambda__783,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_ty(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu2,Uua))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_783
tff(fact_8963_ATP_Olambda__784,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_hk(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,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_784
tff(fact_8964_ATP_Olambda__785,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_yv(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> member(B,aa(A,B,Uu2,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_785
tff(fact_8965_ATP_Olambda__786,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,A),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,A),fun(A,fun(A,$o)),aTP_Lamp_aiu(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu2,Uuc)
        & ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).

% ATP.lambda_786
tff(fact_8966_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B,Uuc: A] : aa(A,A,aa(B,fun(A,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_afj(fun(B,A),fun(A,fun(B,fun(A,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu2,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).

% ATP.lambda_787
tff(fact_8967_ATP_Olambda__788,axiom,
    ! [Uu2: 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_gy(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uuc)),aa(nat,nat,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_788
tff(fact_8968_ATP_Olambda__789,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: 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_gr(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_789
tff(fact_8969_ATP_Olambda__790,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_gu(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_790
tff(fact_8970_ATP_Olambda__791,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu2: 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_ay(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_791
tff(fact_8971_ATP_Olambda__792,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,nat),Uua: fun(B,nat),Uub: A,Uuc: B] : aa(B,nat,aa(A,fun(B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_aik(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu2,Uub)),aa(B,nat,Uua,Uuc)) ).

% ATP.lambda_792
tff(fact_8972_ATP_Olambda__793,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_0(C)
     => ! [Uu2: fun(A,C),Uua: fun(B,C),Uub: A,Uuc: B] : aa(B,C,aa(A,fun(B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_aif(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu2,Uub)),aa(B,C,Uua,Uuc)) ) ).

% ATP.lambda_793
tff(fact_8973_ATP_Olambda__794,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu2: 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_adh(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu2),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu2,Uub)),aa(D,B,Uua,Uuc)) ).

% ATP.lambda_794
tff(fact_8974_ATP_Olambda__795,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,$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,$o)),aTP_Lamp_ahd(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu2,Uub)
        & aa(B,$o,Uua,Uuc) ) ) ).

% ATP.lambda_795
tff(fact_8975_ATP_Olambda__796,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_agf(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( aa(B,A,Uu2,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_796
tff(fact_8976_ATP_Olambda__797,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_uc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),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,Uu2,Uua))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_797
tff(fact_8977_ATP_Olambda__798,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_ua(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),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,Uu2,Uub)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% ATP.lambda_798
tff(fact_8978_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_sz(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),exp(real,aa(A,real,Uu2,Uub))) ) ).

% ATP.lambda_799
tff(fact_8979_ATP_Olambda__800,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_tb(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),cos(real,aa(A,real,Uu2,Uub))) ) ).

% ATP.lambda_800
tff(fact_8980_ATP_Olambda__801,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_to(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),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,Uu2,Uua))) ) ).

% ATP.lambda_801
tff(fact_8981_ATP_Olambda__802,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_sn(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),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,Uu2,Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% ATP.lambda_802
tff(fact_8982_ATP_Olambda__803,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_tk(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu2),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,Uu2,Uub)))) ) ).

% ATP.lambda_803
tff(fact_8983_ATP_Olambda__804,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_sp(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),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,Uu2,Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ) ).

% ATP.lambda_804
tff(fact_8984_ATP_Olambda__805,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_zp(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = 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,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu2,Uub))),aa(A,B,Uua,Uuc))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_805
tff(fact_8985_ATP_Olambda__806,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_zk(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = 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,Uu2,Uuc)),aa(A,B,Uu2,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_806
tff(fact_8986_ATP_Olambda__807,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu2: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_yu(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$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,Uu2,Uuc))),Uub)) ) ) ).

% ATP.lambda_807
tff(fact_8987_ATP_Olambda__808,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_zr(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu2),Uua),Uub),Uuc) = 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_zq(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,Uu2,Uuc)),aa(A,B,Uu2,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_zq(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_zq(A,A)))))) ) ).

% ATP.lambda_808
tff(fact_8988_ATP_Olambda__809,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_zl(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = 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,Uu2,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))) ) ).

% ATP.lambda_809
tff(fact_8989_ATP_Olambda__810,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_zm(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu2),Uua),Uub),Uuc) = 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,Uu2,Uuc)),aa(A,B,Uu2,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))) ) ).

% ATP.lambda_810
tff(fact_8990_ATP_Olambda__811,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_on(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_om(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_811
tff(fact_8991_ATP_Olambda__812,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$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,$o)),aTP_Lamp_ace(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ? [A6: A] :
            ( ( Uub = A6 )
            & ( Uuc = A6 ) )
        | ? [A6: A,B6: A,C4: A] :
            ( ( Uub = A6 )
            & ( Uuc = C4 )
            & aa(A,$o,aa(A,fun(A,$o),Uua,A6),B6)
            & aa(A,$o,aa(A,fun(A,$o),Uu2,B6),C4) ) ) ) ).

% ATP.lambda_812
tff(fact_8992_ATP_Olambda__813,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$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,$o)),aTP_Lamp_acf(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ? [A6: A,B6: A] :
            ( ( Uub = A6 )
            & ( Uuc = B6 )
            & aa(A,$o,aa(A,fun(A,$o),Uu2,A6),B6) )
        | ? [A6: A,B6: A,C4: A] :
            ( ( Uub = A6 )
            & ( Uuc = C4 )
            & aa(A,$o,aa(A,fun(A,$o),Uua,A6),B6)
            & aa(A,$o,aa(A,fun(A,$o),Uu2,B6),C4) ) ) ) ).

% ATP.lambda_813
tff(fact_8993_ATP_Olambda__814,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$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),$o)),aTP_Lamp_aci(fun(A,fun(A,$o)),fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ? [Y5: A,Ys4: list(A)] :
            ( ( Uub = nil(A) )
            & ( Uuc = aa(list(A),list(A),cons(A,Y5),Ys4) ) )
        | ? [X3: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),cons(A,X3),Xs3) )
            & ( Uuc = aa(list(A),list(A),cons(A,Y5),Ys4) )
            & aa(A,$o,aa(A,fun(A,$o),Uu2,X3),Y5) )
        | ? [X3: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),cons(A,X3),Xs3) )
            & ( Uuc = aa(list(A),list(A),cons(A,Y5),Ys4) )
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu2,X3),Y5)
            & ~ aa(A,$o,aa(A,fun(A,$o),Uu2,Y5),X3)
            & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uua,Xs3),Ys4) ) ) ) ).

% ATP.lambda_814
tff(fact_8994_ATP_Olambda__815,axiom,
    ! [A: $tType,Uu2: 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_pi(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu2),Uua),Uub),Uuc) = aa(list(A),list(A),cons(A,Uu2),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_815
tff(fact_8995_ATP_Olambda__816,axiom,
    ! [A: $tType,B: $tType,Uu2: $o,Uua: fun(A,fun(B,$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_cl($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu2)),Uua),Uub),Uuc)
    <=> ( (Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_816
tff(fact_8996_ATP_Olambda__817,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(C,A),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,set(B),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_aim(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu2),Uua),Uub),Uuc) = image(C,B,Uua,aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),vimage(C,A,Uu2,aa(set(A),set(A),insert(A,Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_817
tff(fact_8997_ATP_Olambda__818,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_yz(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$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_ct(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uuc)),set_ord_atMost(nat,Uua)))) ) ) ).

% ATP.lambda_818
tff(fact_8998_ATP_Olambda__819,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_abo(A,fun(list(A),fun(A,fun(list(A),A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),ord_min(A),Uu2),min_list(A,Uua)) ) ).

% ATP.lambda_819
tff(fact_8999_ATP_Olambda__820,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(B,A),Uua: fun(C,fun(list(C),B)),Uub: C,Uuc: list(C)] : aa(list(C),A,aa(C,fun(list(C),A),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_abn(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),Uu2),Uua),Uub),Uuc) = aa(B,A,Uu2,aa(list(C),B,aa(C,fun(list(C),B),Uua,Uub),Uuc)) ).

% ATP.lambda_820
tff(fact_9000_ATP_Olambda__821,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu2: 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_bm(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu2),Uua),Uub),Uuc) = aa(B,A,Uu2,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).

% ATP.lambda_821
tff(fact_9001_ATP_Olambda__822,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_kd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,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_822
tff(fact_9002_ATP_Olambda__823,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_kb(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,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_823
tff(fact_9003_ATP_Olambda__824,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_fy(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,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_824
tff(fact_9004_ATP_Olambda__825,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_bs(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,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_825
tff(fact_9005_ATP_Olambda__826,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,Uu2: fun(product_prod(B,C),A),Uua: fun(D,B),Uub: D,Uuc: C] : aa(C,A,aa(D,fun(C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_adg(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu2,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uua,Uub)),Uuc)) ).

% ATP.lambda_826
tff(fact_9006_ATP_Olambda__827,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Uu2: fun(product_prod(B,C),A),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,A,aa(B,fun(D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_adi(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu2),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu2,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),aa(D,C,Uua,Uuc))) ).

% ATP.lambda_827
tff(fact_9007_ATP_Olambda__828,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,$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),$o))),aTP_Lamp_abl(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu2),Uua),Uub),Uuc) = aa(fun(A,fun(B,$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,$o)))),aTP_Lamp_abk(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu2),Uua),Uub),Uuc)) ).

% ATP.lambda_828
tff(fact_9008_ATP_Olambda__829,axiom,
    ! [A: $tType,B: $tType,Uu2: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$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_abj(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu2),Uua),Uub),Uuc) = aa(fun(A,fun(B,$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,$o)))),aTP_Lamp_abi(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu2),Uua),Uub),Uuc)) ).

% ATP.lambda_829
tff(fact_9009_ATP_Olambda__830,axiom,
    ! [Uu2: 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_jq(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),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),Uu2),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_830
tff(fact_9010_ATP_Olambda__831,axiom,
    ! [Uu2: 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_js(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),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),Uu2),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_831
tff(fact_9011_ATP_Olambda__832,axiom,
    ! [A: $tType,Uu2: fun(A,$o),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_ago(fun(A,$o),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu2),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),takeWhile(A,aa(fun(A,$o),fun(A,$o),comp($o,$o,A,fNot),Uu2),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_832
tff(fact_9012_ATP_Olambda__833,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: 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_tg(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu2),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,Uu2,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua)))) ) ).

% ATP.lambda_833
tff(fact_9013_ATP_Olambda__834,axiom,
    ! [A: $tType,Uu2: 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_qt(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu2),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))),aa(list(A),list(A),cons(A,Uu2),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_834
tff(fact_9014_ATP_Olambda__835,axiom,
    ! [Uu2: 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_jw(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),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),Uu2),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_835
tff(fact_9015_ATP_Olambda__836,axiom,
    ! [Uu2: 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_ju(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu2),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),Uu2),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_836
tff(fact_9016_ATP_Olambda__837,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_aee(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ? [Y5: C] :
          ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y5),Uu2)
          & member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y5),Uuc),Uua) ) ) ).

% ATP.lambda_837
tff(fact_9017_ATP_Olambda__838,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( aa(product_prod(A,B),$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_lu(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),$o))),Uu2),Uua),Uub),Uuc)
    <=> ? [A6: C] :
          ( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A6)),aa(C,B,Uub,A6)) )
          & member(C,A6,Uu2) ) ) ).

% ATP.lambda_838
tff(fact_9018_ATP_Olambda__839,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(A,$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,$o)),aTP_Lamp_an(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),Uu2),Uua),Uub),Uuc)
    <=> ? [X3: A,Y5: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X3),Y5) )
          & aa(A,$o,Uu2,X3)
          & aa(B,$o,Uua,Y5) ) ) ).

% ATP.lambda_839
tff(fact_9019_ATP_Olambda__840,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: 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_aek(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu2),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),Uu2),Uuc)),Uud),Uud) ).

% ATP.lambda_840
tff(fact_9020_ATP_Olambda__841,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu2: 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_aef(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu2),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),Uu2),Uuc)),Uud),Uud) ).

% ATP.lambda_841
tff(fact_9021_ATP_Olambda__842,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_tu(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu2),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_tt(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu2),Uua),Uub),Uuc),Uud)),Uu2) ) ).

% ATP.lambda_842
tff(fact_9022_ATP_Olambda__843,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_hj(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu2),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_hi(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),Uud))) ) ).

% ATP.lambda_843
tff(fact_9023_ATP_Olambda__844,axiom,
    ! [Uu2: 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_si(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu2),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_sh(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),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),Uu2),Uuc)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),Uuc)))))) ).

% ATP.lambda_844
tff(fact_9024_ATP_Olambda__845,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_hl(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu2),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_hk(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud)) ) ).

% ATP.lambda_845
tff(fact_9025_ATP_Olambda__846,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: 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_hh(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu2),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),Uu2),Uud))) ) ).

% ATP.lambda_846
tff(fact_9026_ATP_Olambda__847,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: 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_hd(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu2),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),Uu2)),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),Uu2),Uud))) ) ).

% ATP.lambda_847
tff(fact_9027_ATP_Olambda__848,axiom,
    ! [A: $tType,Uu2: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$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,$o))),aTP_Lamp_acy(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu2),Uua),Uub),Uuc),Uud)
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu2),transitive_trancl(A,Uub))
          | ( Uuc = Uu2 ) )
        & ( 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_848
tff(fact_9028_ATP_Olambda__849,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: 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_he(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu2),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),Uu2),Uud))) ) ).

% ATP.lambda_849
tff(fact_9029_ATP_Olambda__850,axiom,
    ! [A: $tType,Uu2: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$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,$o))),aTP_Lamp_acx(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu2),Uua),Uub),Uuc),Uud)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu2),transitive_rtrancl(A,Uub))
        & 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_850
tff(fact_9030_ATP_Olambda__851,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: 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_hi(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu2),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,Uu2,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_851
tff(fact_9031_ATP_Olambda__852,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: 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_tm(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu2),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,Uu2,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),one_one(nat)))) ) ).

% ATP.lambda_852
tff(fact_9032_ATP_Olambda__853,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_agw(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),Uu2),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),aa(int,A,ring_1_of_int(A),Uuc)),power_int(A,aa(B,A,Uu2,Uua),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uuc),one_one(int))))) ) ).

% ATP.lambda_853
tff(fact_9033_ATP_Olambda__854,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_0(C)
     => ! [Uu2: fun(A,C),Uua: fun(B,C),Uub: set(A),Uuc: set(B),Uud: C] :
          ( aa(C,$o,aa(set(B),fun(C,$o),aa(set(A),fun(set(B),fun(C,$o)),aa(fun(B,C),fun(set(A),fun(set(B),fun(C,$o))),aTP_Lamp_aig(fun(A,C),fun(fun(B,C),fun(set(A),fun(set(B),fun(C,$o)))),Uu2),Uua),Uub),Uuc),Uud)
        <=> ? [A6: A,B6: B] :
              ( ( Uud = aa(C,C,aa(C,fun(C,C),times_times(C),aa(A,C,Uu2,A6)),aa(B,C,Uua,B6)) )
              & member(A,A6,Uub)
              & member(B,B6,Uuc) ) ) ) ).

% ATP.lambda_854
tff(fact_9034_ATP_Olambda__855,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_tt(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu2),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_tr(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)),Uu2),aa(set(A),set(A),insert(A,Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_855
tff(fact_9035_ATP_Olambda__856,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: 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_td(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = 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,Uu2,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_856
tff(fact_9036_ATP_Olambda__857,axiom,
    ! [A: $tType,B: $tType,Uu2: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$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_abi(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu2),Uua),Uub),Uuc),Uud),Uue)
    <=> ( 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),Uu2)
        | ( ( Uub = Uud )
          & 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_857
tff(fact_9037_ATP_Olambda__858,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: 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_tw(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu2,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,Uu2,Uub)))),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,Uu2,Uub)))) ) ).

% ATP.lambda_858
tff(fact_9038_ATP_Olambda__859,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: 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_sr(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),real_V8093663219630862766scaleR(B,aa(A,real,Uu2,Uub),aa(A,B,Uud,Uue))),real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_859
tff(fact_9039_ATP_Olambda__860,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu2: 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_sx(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),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,Uu2,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_860
tff(fact_9040_ATP_Olambda__861,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: 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_tq(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),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,Uu2,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))))),divide_divide(B,aa(A,B,Uua,Uue),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_861
tff(fact_9041_ATP_Olambda__862,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,$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,$o)))),aTP_Lamp_abk(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu2),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa(A,$o,Uu2,Uud)
        & 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_862
tff(fact_9042_ATP_Olambda__863,axiom,
    ! [Uu2: 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_mt(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Uu2),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uue),Uuf)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuf),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu2))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uud))
           => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I3)),X_12)
            <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Uub),I3) ) )
        & $ite(
            Uue = Uuf,
            ! [X3: vEBT_VEBT] :
              ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X7) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu2))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X3)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),X3)
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_863
tff(fact_9043_ATP_Olambda__864,axiom,
    ! [B: $tType,A: $tType,Uu2: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_qr($o,fun(A,fun(B,$o)),(Uu2)),Uua),Uub)
    <=> (Uu2) ) ).

% ATP.lambda_864
tff(fact_9044_ATP_Olambda__865,axiom,
    ! [C: $tType,D: $tType,Uu2: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_aho(set(D),fun(C,set(D)),Uu2),Uua) = Uu2 ).

% ATP.lambda_865
tff(fact_9045_ATP_Olambda__866,axiom,
    ! [B: $tType,C: $tType,Uu2: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_ain(set(C),fun(B,set(C)),Uu2),Uua) = Uu2 ).

% ATP.lambda_866
tff(fact_9046_ATP_Olambda__867,axiom,
    ! [A: $tType,B: $tType,Uu2: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_ahf(set(B),fun(A,set(B)),Uu2),Uua) = Uu2 ).

% ATP.lambda_867
tff(fact_9047_ATP_Olambda__868,axiom,
    ! [A: $tType,Uu2: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_ahb(set(A),fun(list(A),set(A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_868
tff(fact_9048_ATP_Olambda__869,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_ahj(set(A),fun(B,set(A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_869
tff(fact_9049_ATP_Olambda__870,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_ahu(set(A),fun(A,set(A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_870
tff(fact_9050_ATP_Olambda__871,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_py(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_871
tff(fact_9051_ATP_Olambda__872,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_ku(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_872
tff(fact_9052_ATP_Olambda__873,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_kv(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_873
tff(fact_9053_ATP_Olambda__874,axiom,
    ! [A: $tType,Uu2: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_adk(A,fun(list(A),A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_874
tff(fact_9054_ATP_Olambda__875,axiom,
    ! [A: $tType,Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_aem(A,fun(nat,A),Uu2),Uua) = Uu2 ).

% ATP.lambda_875
tff(fact_9055_ATP_Olambda__876,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_lp(A,fun(B,A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_876
tff(fact_9056_ATP_Olambda__877,axiom,
    ! [B: $tType,A: $tType,Uu2: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_lo(B,fun(A,A)),Uu2),Uua) = Uua ).

% ATP.lambda_877
tff(fact_9057_ATP_Olambda__878,axiom,
    ! [A: $tType,Uu2: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_afo(A,fun(list(A),list(A))),Uu2),Uua) = Uua ).

% ATP.lambda_878
tff(fact_9058_ATP_Olambda__879,axiom,
    ! [A: $tType,Uu2: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_abt(A,fun(list(A),$o)),Uu2),Uua)
    <=> $false ) ).

% ATP.lambda_879
tff(fact_9059_ATP_Olambda__880,axiom,
    ! [A: $tType,Uu2: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_abs(A,fun(list(A),$o)),Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_880
tff(fact_9060_ATP_Olambda__881,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_cm(A,fun(B,$o)),Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_881
tff(fact_9061_ATP_Olambda__882,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agj(A,fun(A,$o)),Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_882
tff(fact_9062_ATP_Olambda__883,axiom,
    ! [Uu2: complex] : aa(complex,complex,aTP_Lamp_bt(complex,complex),Uu2) = Uu2 ).

% ATP.lambda_883
tff(fact_9063_ATP_Olambda__884,axiom,
    ! [Uu2: nat] : aa(nat,nat,aTP_Lamp_ca(nat,nat),Uu2) = Uu2 ).

% ATP.lambda_884
tff(fact_9064_ATP_Olambda__885,axiom,
    ! [Uu2: int] : aa(int,int,aTP_Lamp_ax(int,int),Uu2) = Uu2 ).

% ATP.lambda_885
tff(fact_9065_ATP_Olambda__886,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_zq(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_886
tff(fact_9066_ATP_Olambda__887,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_rd(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_887
tff(fact_9067_ATP_Olambda__888,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_zw(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_888
tff(fact_9068_ATP_Olambda__889,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_afz(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_889
tff(fact_9069_ATP_Olambda__890,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_ms(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_890
tff(fact_9070_ATP_Olambda__891,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_ab(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_891
tff(fact_9071_ATP_Olambda__892,axiom,
    ! [A: $tType,Uu2: A] : aa(A,A,aTP_Lamp_nt(A,A),Uu2) = Uu2 ).

% ATP.lambda_892
tff(fact_9072_ATP_Olambda__893,axiom,
    ! [A: $tType,B: $tType,Uu2: A] : aa(A,set(B),aTP_Lamp_ahh(A,set(B)),Uu2) = top_top(set(B)) ).

% ATP.lambda_893
tff(fact_9073_ATP_Olambda__894,axiom,
    ! [A: $tType,B: $tType,Uu2: A] : aa(A,set(B),aTP_Lamp_ahi(A,set(B)),Uu2) = bot_bot(set(B)) ).

% ATP.lambda_894
tff(fact_9074_ATP_Olambda__895,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_ae(A,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_895
tff(fact_9075_ATP_Olambda__896,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_ff(B,A),Uu2) = one_one(A) ) ).

% ATP.lambda_896
tff(fact_9076_ATP_Olambda__897,axiom,
    ! [A: $tType,Uu2: A] : aa(A,real,aTP_Lamp_ld(A,real),Uu2) = one_one(real) ).

% ATP.lambda_897
tff(fact_9077_ATP_Olambda__898,axiom,
    ! [A: $tType,Uu2: A] : aa(A,nat,aTP_Lamp_kw(A,nat),Uu2) = one_one(nat) ).

% ATP.lambda_898
tff(fact_9078_ATP_Olambda__899,axiom,
    ! [B: $tType,A: $tType,Uu2: B] : aa(B,option(A),aTP_Lamp_acn(B,option(A)),Uu2) = none(A) ).

% ATP.lambda_899
tff(fact_9079_ATP_Olambda__900,axiom,
    ! [A: $tType,C: $tType,Uu2: A] : aa(A,option(C),aTP_Lamp_nv(A,option(C)),Uu2) = none(C) ).

% ATP.lambda_900
tff(fact_9080_ATP_Olambda__901,axiom,
    ! [A: $tType,B: $tType,Uu2: A] : aa(A,option(B),aTP_Lamp_ph(A,option(B)),Uu2) = none(B) ).

% ATP.lambda_901
tff(fact_9081_ATP_Olambda__902,axiom,
    ! [Uu2: nat] :
      ( aa(nat,$o,aTP_Lamp_lj(nat,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_902
tff(fact_9082_ATP_Olambda__903,axiom,
    ! [A: $tType,Uu2: A] :
      ( aa(A,$o,aTP_Lamp_mv(A,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_903
tff(fact_9083_ATP_Olambda__904,axiom,
    ! [Uu2: nat] :
      ( aa(nat,$o,aTP_Lamp_li(nat,$o),Uu2)
    <=> $true ) ).

% ATP.lambda_904
tff(fact_9084_ATP_Olambda__905,axiom,
    ! [A: $tType,Uu2: A] :
      ( aa(A,$o,aTP_Lamp_mu(A,$o),Uu2)
    <=> $true ) ).

% ATP.lambda_905
tff(fact_9085_ATP_Olambda__906,axiom,
    ! [A: $tType,Uu2: A] : aa(A,fun(nat,nat),aTP_Lamp_op(A,fun(nat,nat)),Uu2) = suc ).

% ATP.lambda_906

% Type constructors (784)
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,
    ! [A11: $tType] : bounded_lattice(filter(A11)) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_3,axiom,
    bounded_lattice($o) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_4,axiom,
    ! [A11: $tType] : bounded_lattice(set(A11)) ).

tff(tcon_fun___Lattices_Obounded__lattice_5,axiom,
    ! [A11: $tType,A14: $tType] :
      ( bounded_lattice(A14)
     => bounded_lattice(fun(A11,A14)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A11: $tType,A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => condit1219197933456340205attice(fun(A11,A14)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A11: $tType,A14: $tType] :
      ( counta3822494911875563373attice(A14)
     => counta3822494911875563373attice(fun(A11,A14)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A11: $tType,A14: $tType] :
      ( comple592849572758109894attice(A14)
     => comple592849572758109894attice(fun(A11,A14)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A11: $tType,A14: $tType] :
      ( bounded_lattice(A14)
     => bounde4967611905675639751up_bot(fun(A11,A14)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A11: $tType,A14: $tType] :
      ( bounded_lattice(A14)
     => bounde4346867609351753570nf_top(fun(A11,A14)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A11: $tType,A14: $tType] :
      ( comple6319245703460814977attice(A14)
     => comple6319245703460814977attice(fun(A11,A14)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A11: $tType,A14: $tType] :
      ( boolea8198339166811842893lgebra(A14)
     => boolea8198339166811842893lgebra(fun(A11,A14)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A11: $tType,A14: $tType] :
      ( semilattice_sup(A14)
     => semilattice_sup(fun(A11,A14)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A11: $tType,A14: $tType] :
      ( semilattice_inf(A14)
     => semilattice_inf(fun(A11,A14)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A11: $tType,A14: $tType] :
      ( distrib_lattice(A14)
     => distrib_lattice(fun(A11,A14)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A11: $tType,A14: $tType] :
      ( order_top(A14)
     => order_top(fun(A11,A14)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A11: $tType,A14: $tType] :
      ( order_bot(A14)
     => order_bot(fun(A11,A14)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A11: $tType,A14: $tType] :
      ( preorder(A14)
     => preorder(fun(A11,A14)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A11: $tType,A14: $tType] :
      ( lattice(A14)
     => lattice(fun(A11,A14)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A11: $tType,A14: $tType] :
      ( order(A14)
     => order(fun(A11,A14)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A11: $tType,A14: $tType] :
      ( ord(A14)
     => ord(fun(A11,A14)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A11: $tType,A14: $tType] :
      ( uminus(A14)
     => uminus(fun(A11,A14)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A11: $tType,A14: $tType] :
      ( minus(A14)
     => minus(fun(A11,A14)) ) ).

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___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___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___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_10,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_11,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_12,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_13,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_14,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_15,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_16,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_17,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_18,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_19,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_20,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_21,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_22,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_23,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_24,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_25,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_26,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_27,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_28,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_29,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_30,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_31,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_32,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_33,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_34,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_35,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_36,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_37,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_38,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_39,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_40,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_41,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_42,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_43,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_44,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_45,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_46,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_47,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_48,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_49,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_50,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_51,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_52,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_53,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_54,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_55,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_56,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_57,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_58,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_59,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_60,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_61,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_62,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_63,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_64,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_65,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_66,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_67,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_68,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_69,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_70,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_71,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_72,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_73,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_74,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_75,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_76,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_77,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_78,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_79,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_80,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_81,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_82,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_83,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_84,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_85,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_86,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_87,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_88,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_89,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_90,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_91,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_92,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_93,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_94,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_95,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_96,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_97,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_98,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_99,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_100,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_101,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_102,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_103,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_104,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_105,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_106,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_107,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_108,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_109,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_110,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_111,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_112,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_113,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_114,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_115,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_116,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_117,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_118,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_119,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_120,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_121,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_122,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_123,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_124,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_125,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_126,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_127,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_128,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_129,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_130,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_131,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_132,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_133,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_134,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_135,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_136,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_137,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_138,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_139,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_140,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_141,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_142,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_143,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_144,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_145,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_146,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_147,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_148,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_149,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_150,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_151,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_152,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_153,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_154,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_155,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_156,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_157,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_158,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_159,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_160,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_161,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_162,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_163,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_164,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_165,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_166,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_167,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_168,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_169,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_170,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_171,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_172,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_173,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_174,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_175,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_176,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_177,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_178,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_179,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_180,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_181,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_182,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_183,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_184,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_185,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_186,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_187,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_188,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_189,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_190,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_191,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_192,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_193,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_194,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_195,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_196,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_197,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_198,axiom,
    ! [A11: $tType] : condit1219197933456340205attice(set(A11)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_199,axiom,
    ! [A11: $tType] : counta3822494911875563373attice(set(A11)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_200,axiom,
    ! [A11: $tType] : comple592849572758109894attice(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_201,axiom,
    ! [A11: $tType] : bounde4967611905675639751up_bot(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_202,axiom,
    ! [A11: $tType] : bounde4346867609351753570nf_top(set(A11)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_203,axiom,
    ! [A11: $tType] : comple6319245703460814977attice(set(A11)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_204,axiom,
    ! [A11: $tType] : boolea8198339166811842893lgebra(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_205,axiom,
    ! [A11: $tType] : semilattice_sup(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_206,axiom,
    ! [A11: $tType] : semilattice_inf(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_207,axiom,
    ! [A11: $tType] : distrib_lattice(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_208,axiom,
    ! [A11: $tType] : order_top(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_209,axiom,
    ! [A11: $tType] : order_bot(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_210,axiom,
    ! [A11: $tType] : preorder(set(A11)) ).

tff(tcon_Set_Oset___Lattices_Olattice_211,axiom,
    ! [A11: $tType] : lattice(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder_212,axiom,
    ! [A11: $tType] : order(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oord_213,axiom,
    ! [A11: $tType] : ord(set(A11)) ).

tff(tcon_Set_Oset___Groups_Ouminus_214,axiom,
    ! [A11: $tType] : uminus(set(A11)) ).

tff(tcon_Set_Oset___Groups_Ominus_215,axiom,
    ! [A11: $tType] : minus(set(A11)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_216,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_217,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_218,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_219,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_220,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_221,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_222,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_223,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_224,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_225,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_226,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_227,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_228,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_229,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_230,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_231,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_232,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_233,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_234,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_235,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_236,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_237,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_238,axiom,
    minus($o) ).

tff(tcon_List_Olist___Nat_Osize_239,axiom,
    ! [A11: $tType] : size(list(A11)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_240,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_241,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_242,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_243,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_244,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_245,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_246,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_247,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_248,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_249,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_250,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_251,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_252,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_253,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_254,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_255,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_256,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_257,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_258,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_259,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_260,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_261,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_262,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_263,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_264,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_265,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_266,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_267,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_268,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_269,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_270,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_271,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_272,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_273,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_274,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_275,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_276,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_277,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_278,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_279,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_280,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_281,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_282,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_283,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_284,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_285,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_286,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_287,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_288,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_289,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_290,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_291,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_292,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_293,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_294,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_295,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_296,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_297,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_298,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_299,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_300,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_301,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_302,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_303,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_304,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_305,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_306,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_307,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_308,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_309,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_310,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_311,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_312,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_313,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_314,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_315,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_316,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_317,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_318,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_319,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_320,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_321,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_322,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_323,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_324,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_325,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_326,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_327,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_328,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_329,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_330,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_331,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_332,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_333,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_334,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_335,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_336,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_337,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_338,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_339,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_340,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_341,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_342,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_343,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_344,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_345,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_346,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Nat_Osize_347,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_348,axiom,
    ! [A11: $tType,A14: $tType] : size(sum_sum(A11,A14)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_349,axiom,
    ! [A11: $tType] : condit1219197933456340205attice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_350,axiom,
    ! [A11: $tType] : counta3822494911875563373attice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_351,axiom,
    ! [A11: $tType] : bounde4967611905675639751up_bot(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_352,axiom,
    ! [A11: $tType] : bounde4346867609351753570nf_top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_353,axiom,
    ! [A11: $tType] : comple6319245703460814977attice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_354,axiom,
    ! [A11: $tType] : semilattice_sup(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_355,axiom,
    ! [A11: $tType] : semilattice_inf(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_356,axiom,
    ! [A11: $tType] : distrib_lattice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_357,axiom,
    ! [A11: $tType] : order_top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_358,axiom,
    ! [A11: $tType] : order_bot(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_359,axiom,
    ! [A11: $tType] : preorder(filter(A11)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_360,axiom,
    ! [A11: $tType] : lattice(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_361,axiom,
    ! [A11: $tType] : order(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_362,axiom,
    ! [A11: $tType] : ord(filter(A11)) ).

tff(tcon_Option_Ooption___Nat_Osize_363,axiom,
    ! [A11: $tType] : size(option(A11)) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_364,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_365,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_366,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_367,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_368,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_369,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_370,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_371,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_372,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_373,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_374,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_375,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_376,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_377,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_378,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_379,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_380,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_381,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_382,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_383,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_384,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_385,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_386,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_387,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_388,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_389,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Odist__norm_390,axiom,
    real_V6936659425649961206t_norm(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_391,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_392,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_393,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_394,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_395,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_396,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_397,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_398,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_399,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_400,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_401,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_402,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_403,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_404,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_405,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_406,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_407,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_408,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_409,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_410,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_411,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_412,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_413,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_414,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_415,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_416,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_417,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_418,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_419,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_420,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_421,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_422,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_423,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_424,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_425,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_426,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_427,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_428,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_429,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_430,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_431,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_432,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_433,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_434,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_435,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_436,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_437,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_438,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_439,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_440,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_441,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_442,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_443,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_444,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_445,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_446,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_447,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_448,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_449,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_450,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_451,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_452,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_453,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_454,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_455,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_456,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_457,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_458,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_459,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_460,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_461,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_462,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_463,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_464,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_465,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_466,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_467,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_468,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_469,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_470,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_471,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_472,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_473,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_474,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_475,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_476,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_477,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_478,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_479,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_480,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_481,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_482,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_483,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_484,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_485,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_486,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_487,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_488,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_489,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_490,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_491,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_492,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_493,axiom,
    ! [A11: $tType,A14: $tType] :
      ( ( topolo4958980785337419405_space(A11)
        & topolo4958980785337419405_space(A14) )
     => topolo4958980785337419405_space(product_prod(A11,A14)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_494,axiom,
    ! [A11: $tType,A14: $tType] :
      ( ( topological_t2_space(A11)
        & topological_t2_space(A14) )
     => topological_t2_space(product_prod(A11,A14)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_495,axiom,
    ! [A11: $tType,A14: $tType] : size(product_prod(A11,A14)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_496,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_497,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_498,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_499,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_500,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_501,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_502,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_503,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_504,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_505,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_506,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_507,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_508,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_509,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_510,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_511,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_512,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_513,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_514,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_515,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_516,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_517,axiom,
    minus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_518,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_519,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_520,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_521,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_522,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_523,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_524,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_525,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_526,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_527,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_528,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_529,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_530,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_531,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_532,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_533,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_534,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_535,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_536,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_537,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_538,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_539,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_540,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_541,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_542,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_543,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_544,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_545,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_546,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_547,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_548,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_549,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_550,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_551,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_552,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_553,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_554,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_555,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_556,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_557,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_558,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_559,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_560,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_561,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_562,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_563,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_564,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_565,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_566,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_567,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_568,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_569,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_570,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_571,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_572,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_573,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_574,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_575,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_576,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_577,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_578,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_579,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_580,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_581,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_582,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_583,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_584,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_585,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_586,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_587,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_588,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_589,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_590,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_591,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_592,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_593,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_594,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_595,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_596,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_597,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_598,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_599,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_600,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_601,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_602,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_603,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_604,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_605,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_606,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_607,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_608,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_609,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_610,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_611,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_612,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_613,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_614,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_615,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_616,axiom,
    dvd(code_integer) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_617,axiom,
    size(vEBT_VEBT) ).

% Helper facts (4)
tff(help_fNot_2_1_U,axiom,
    ! [P: $o] :
      ( (P)
      | aa($o,$o,fNot,(P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: $o] :
      ( ~ aa($o,$o,fNot,(P))
      | ~ (P) ) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X9: A,Y: A] :
      ( ( X9 != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X9),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X9: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X9),Y)
      | ( X9 = Y ) ) ).

% Free types (1)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

% Conjectures (1)
tff(conj_0,conjecture,
    ( ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary),xa) = aa(nat,option(nat),some(nat),sxa) )
  <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),mi),ma)),deg,treeList,summary)),xa,sxa) ) ).

%------------------------------------------------------------------------------